Local predictability in a simple model of atmospheric balance

International audienceThe 2 degree-of-freedom elastic pendulum equations can be considered as the lowest order analogue of interacting low-frequency (slow) Rossby-Haurwitz and high-frequency (fast) gravity waves in the atmosphere. The strength of the coupling between the low and the high frequency waves is controlled by a single coupling parameter, e, defined by the ratio of the fast and slow characteristic time scales. In this paper, efficient, high accuracy, and symplectic structure preserving numerical solutions are designed for the elastic pendulum equation in order to study the role balanced dynamics play in local predictability. To quantify changes in the local predictability, two measures are considered: the local Lyapunov number and the leading singular value of the tangent linear map. It is shown, both based on theoretical considerations and numerical experiments, that there exist regions of the phase space where the local Lyapunov number indicates exceptionally high predictability, while the dominant singular value indicates exceptionally low predictability. It is also demonstrated that the local Lyapunov number has a tendency to choose instabilities associated with balanced motions, while the dominant singular value favors instabilities related to highly unbalanced motions. The implications of these findings for atmospheric dynamics are also discussed

Comparison of Entropy Production Rates in Two Different Types of Self-organized Flows: B\'{e}nard Convection and Zonal flow

Entropy production rate (EPR) is often effective to describe how a structure is self-organized in a nonequilibrium thermodynamic system. The "minimum EPR principle" is widely applicable to characterizing self-organized structures, but is sometimes disproved by observations of "maximum EPR states." Here we delineate a dual relation between the minimum and maximum principles; the mathematical representation of the duality is given by a Legendre transformation. For explicit formulation, we consider heat transport in the boundary layer of fusion plasma [Phys. Plasmas {\bf 15}, 032307 (2008)]. The mechanism of bifurcation and hysteresis (which are the determining characteristics of the so-called H-mode, a self-organized state of reduced thermal conduction) is explained by multiple tangent lines to a pleated graph of an appropriate thermodynamic potential. In the nonlinear regime, we have to generalize Onsager's dissipation function. The generalized function is no longer equivalent to EPR; then EPR ceases to be the determinant of the operating point, and may take either minimum or maximum values depending on how the system is driven

A superadditivity and submultiplicativity property for cardinalities of sumsets

For finite sets of integers A1, . . . ,An we study the cardinality of the n-fold sumset A1 + · · · + An compared to those of (n − 1)-fold sumsets A1 + · · · + Ai−1 + Ai+1 + · · · + An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets

Relaxation in homogeneous and non-homogeneous polarized systems. A mesoscopic entropy approach

The dynamics of a degree of freedom associated to an axial vector in contact with a heat bath is decribed by means of a probability distribution function obeying a Fokker-Planck equation. The equation is derived by using mesoscopic non-equilibrium thermodynamics and permits a formulation of a dynamical theory for the axial degree of freedom (orientation, polarization) and its associated order parameter. The theory is used to describe dielectric relaxation in homogeneous and non-homogeneous systems in the presence of strong electric fields. In the homogeneous case, we obtain the dependence of the relaxation time on the external field as observed in experiments. In the non-homogeneous case, our model account for the two observed maxima of the dielectric loss giving a good quantitative description of experimental data at all frequencies, especially for systems with low molecular mass.Comment: 19 pages, 3 table

Identification of an average temperature and a dynamical pressure in a multitemperature mixture of fluids

We present a classical approach of a mixture of compressible fluids when each constituent has its own temperature. The introduction of an average temperature together with the entropy principle dictates the classical Fick law for diffusion and also novel constitutive equations associated with the difference of temperatures between the components. The constitutive equations fit with results recently obtained through Maxwellian iteration procedure in extended thermodynamics theory of multitemperature mixtures. The differences of temperatures between the constituents imply the existence of a new dynamical pressure even if the fluids have a zero bulk viscosity. The nonequilibrium dynamical pressure can be measured and may be convenient in several physical situations as for example in cosmological circumstances where - as many authors assert - a dynamical pressure played a major role in the evolution of the early universe.Comment: 16 page

Thermodynamic Field Theory with the Iso-Entropic Formalism

A new formulation of the thermodynamic field theory (TFT) is presented. In this new version, one of the basic restriction in the old theory, namely a closed-form solution for the thermodynamic field strength, has been removed. In addition, the general covariance principle is replaced by Prigogine's thermodynamic covariance principle (TCP). The introduction of TCP required the application of an appropriate mathematical formalism, which has been referred to as the iso-entropic formalism. The validity of the Glansdorff-Prigogine Universal Criterion of Evolution, via geometrical arguments, is proven. A new set of thermodynamic field equations, able to determine the nonlinear corrections to the linear ("Onsager") transport coefficients, is also derived. The geometry of the thermodynamic space is non-Riemannian tending to be Riemannian for hight values of the entropy production. In this limit, we obtain again the same thermodynamic field equations found by the old theory. Applications of the theory, such as transport in magnetically confined plasmas, materials submitted to temperature and electric potential gradients or to unimolecular triangular chemical reactions can be found at references cited herein.Comment: 35 page

Gamow Shell Model Description of Weakly Bound Nuclei and Unbound Nuclear States

We present the study of weakly bound, neutron-rich nuclei using the nuclear shell model employing the complex Berggren ensemble representing the bound single-particle states, unbound Gamow states, and the non-resonant continuum. In the proposed Gamow Shell Model, the Hamiltonian consists of a one-body finite depth (Woods-Saxon) potential and a residual two-body interaction. We discuss the basic ingredients of the Gamow Shell Model. The formalism is illustrated by calculations involving {\it several} valence neutrons outside the double-magic core: $^{6-10}$He and $^{18-22}$O.Comment: 19 pages, 20 encapsulated PostScript figure