As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, we study in this article the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV2s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV2s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the earth's climate. From the mathematical point of view, by integrating the diagnostic equations we present two new formulations of the PEs and the PEV2s. Then we establish some mathematical settings for all the three systems, and obtain the existence and time-analyticity of solutions to the equations. Then we establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems. From the modelling point of view, the PEV2s are introduced in this paper for the first time. Even though the PEs have been studied for a period of time, the new formulation of the PEs is introduced here for the first time. These new formulations of both the PEs and the PEV2S play a crucial role not only for the mathematical studies in this paper, but for the further numerical analysis, which will be developed elsewhere.
We judge symplectic integrators by the accuracy with which they represent the Hamiltonian function. This accuracy is computed, compared and tested for several different methods. We develop new, highly accurate explicit fourth- and fifth-order methods valid when the Hamiltonian is separable with quadratic kinetic energy. For the near-integrable case, we confirm several of their properties expected from KAM theory; convergence of some of the characteristics of chaotic motions are also demonstrated. We point out cases in which long-time stability is intrinsically lost.
The primitive equations are the fundamental equations of atmospheric dynamics. With the purpose of understanding the mechanism of long-term weather prediction and climate changes, we study in this paper as a first step towards this long-range project what is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations we present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. We then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems.
The solution of a stationary k-dimensional Schrodinger equation H-PSI = E-PSI in the semiclassical limit HBAR > 0 is reduced to a discrete (k - 1)-dimensional quantum map psi = T-psi where the integral kernel (the matrix) T is built through classical trajectories corresponding to the classical Poincare map of the given problem. High-excited energy eigenvalues obey the quantization condition zeta(s)(E) = 0 where the function zeta(s)(E) = det(1 - T) coincides with the Selberg zeta function defined as the product over primitive per-iodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.
We derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Our approach is based on the fact that the billiard boundary defines two dual problems: the `inside problem' of the bounded dynamics, and the `outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. We obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. We compare our result to secular equations which were recently derived by other means, and provide some numerical data which illustrate our method when applied to the quantization of the Sinai billiard.
Weakly nonlinear gravity waves of given wavenumber in a horizontally unbounded two-dimensional domain are expected to undergo slow modulations in space and time. Together with an attendant analysis of the water wave equations, this paper gives a mathematical justification of the modulation approximation. It proves that the resulting wavepacket, whose envelope is governed by the cubic nonlinear Schrodinger equation, is a solution of the water wave equations to leading order. An upper bound of the remainder is also provided.
We introduce a class of analytic hyperbolic maps and prove that the time correlation functions associated with analytic observables have a well-defined spectrum satisfying exponential bounds. From the stability of the fixed points of the iterated map, we construct a Fredholm determinant which is an entire function of a complex variable and we show that from its roots we can calculate the spectral values. This paper extends previously obtained results for purely expanding analytic maps and analytic observables as well as for C1+epsilon Axiom A diffeomorphisms and Holder continuous observables. It gives a new, improved, approach to the case of real analytic Axiom A systems and analytic observables.
Greene proposed a relationship between existence of an invariant circle for an area-preserving map and the 'residues' of periodic orbits with nearby rotation number. It remains without a doubt the most effective practical criterion for calculating the breakup of invariant cricles. In this paper four results are proved which go a long way towards placing the criterion on a rigorous foundation, and the issues remaining to be resolved in order to formulate a rigorous version of the criterion are discussed.
In this paper, new dimensional reductions and exact solutions for a generalized nonlinear Schrodinger equation are presented. These are obtained using an extension of the direct method, originally developed by Clarkson and Kruskal to study dimensional reductions of the Boussinesq equation, which involves no group theoretical techniques.
We consider dynamical systems generated by time-dependent periodic Lagrangians on a closed manifold M. An invariant probability mu of such a system has an homology rho(mu) is-an-element-of H-1(M, R) describing (roughly speaking) the average homological position of mu a.e. orbit, and an action A(mu) defined as the integral with respect to mu of the Lagrangian. The minimizing measures are defined as the invariant probabilities of the system that minimize the action among those that have a given homology-gamma. Their action beta(gamma) define a convex function-beta : H-1(M, R) > R. These concepts were introduced by Mather where he proved several theorems about them. In this paper we further develop Mather's theory, giving a characterization of minimizing measures and proving properties about minimizing measures whose homologies are strictly extremal points of the function-beta.
We prove the existence of 'cantori' of all incommensurate rotation vectors, for symplectic maps of arbitrary dimension near enough to any non-degenerate anti-integrable limit, and derive an asymptotic form for them. Cantori are invariant Cantor sets which can be though of as remnants of KAM tori.
We consider the steady group motions of a rigid body with a fixed point moving in a gravitational field. For an asymmetric top, rotation about the axis of gravity is the only permissible group motion; for a Lagrange top, simultaneous rotation about the axis of gravity and spin about the axis of symmetry of the top is permissible. Our analysis of the heavy top follows the reduced energy momentum method of Simo et al, which is applicable to a wide range of conservative systems with symmetry. Steady group motions are characterized as solutions of a variational problem on the configuration space; local minima of the amended potential correspond to nonlinearly orbitally stable steady motions. The combination of a low-dimensional configuration space and a relatively large number of parameters that produce substantial qualitative changes in the dynamics makes possible a thorough, detailed analysis, which not only reproduces the classical results for this well known system, but leads to some results which we believe are new. We determine general equilibrium and nonlinear stability conditions for steady group motions of a heavy top with a fixed point. We rederive the classical equilibrium and stability conditions for sleeping tops and precessing Lagrange tops, analyse in detail the stability of a family of steady rotations of tilted tops which bifurcate from the branch of sleeping tops parametrized by angular velocity, and classify the possible stability transitions of an arbitrary top as its angular velocity is increased. We obtain a simple, general expression for the characteristic polynomial of the linearized equations of motion and analyse the linear stability of both sleeping tops and the family of tilted top motions previously mentioned. Finally, we demonstrate the coexistence of stable branches of steadily precessing tops that bifurcate from the branch of sleeping Lagrange tops throughout the range of angular velocities for which the sleeping top is stable.
We review results concerning normal forms, fronts, pattern formation, and convergence for certain one-dimensional partial differential equations. The relationship between these results is put into perspective and is supplemented by a series of unpublished results on the convergence problem for parabolic PDEs. The paper does not require special prerequisites and we hope to interest the non-specialist in this nice subject.
In this paper we examine the dependence of the energy levels of a classically chaotic system on a parameter. We present numerical results which justify the use of a random matrix model for the statistical properties of this dependence. We illustrate the application of our model by calculating both the number of avoided crossings as a function of gap size and the distribution of curvatures of energy levels for a chaotic billiard: the distribution of large curvatures is determined by the density of avoided crossings. Our results confirm that the matrix elements are Gaussian distributed in the semiclassical limit, but we characterize significant deviations from the Gaussian distribution at finite energies.
The relations between equilibrium state, Gibbs state, and eigenvector of the (adjoint) transfer operator are described for maps satisfying positive expansiveness and specification. In particular we show how the variational principle defining an equilibrium state can be converted into eigenvalue equations for the transfer operator and its adjoint. The results presented here are largely based on the work of N T A Haydn and the author, relating equilibrium and Gibbs state for homeomorphisms satisfying expansiveness and specification.
We present a detailed bifurcation analysis for the travelling-wave solutions of the Kuramoto-Sivashinsky equation, with an emphasis on periodic solutions. The solutions are described by a 1-parameter, reversible third-order ODE. In two previous papers we described new aspects in the observed bifurcations: the 'noose' bifurcation, and a novel kind of 'Shil'nikov' behaviour. This paper brings everything together, and considers the one remaining new aspect, the connected set of period-multiplying k-bifurcations. We offer a possible explanation for this set by considering a 2-parameter, reversible fourth-order ODE that contains the travelling-wave ODE in a particular limit. It is conjectured that the connected set arises from 1 : n resonances of the eigenvalues of a fixed point.