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We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an -regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of -regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations.
In this paper a technique to obtain a first approximation for singular inverse Sturm-Liouville problems with a symmetrical potential is introduced. The singularity, as a result of unbounded domain (-infinity, infinity), is treated by considering numerically the asymptotic limit of the associated problem on a finite interval (-L, L). In spite of this treatment, the problem has still an ill-conditioned structure unlike the classical regular ones and needs regularization techniques. Direct computation of eigenvalues in iterative solution procedure is made by means of pseudospectral methods. A fairly detailed description of the numerical algorithm and its applications to specific examples are presented to illustrate the accuracy and convergence behaviour of the proposed approach.
Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
(2015)
The Dirichlet problem for the heat equation in a bounded domain aS, a"e (n+1) is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane t = c, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii's paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.
We study systematically the estimation of Earth's core angular momentum (CAM) variation between 1962.0 and 2008.0 by using core surface flow models derived from the recent geomagnetic field model C(3)FM2. Various flow models are derived by changing four parameters that control the least squares flow inversion. The parameters include the spherical harmonic (SH) truncation degree of the flow models and two Lagrange multipliers that control the weights of two additional constraints. The first constraint forces the energy spectrum of the flow solution to follow a power law l-p, where l is the SH degree and p is the fourth parameter. The second allows to modulate the solution continuously between the dynamical states of tangential geostrophy (TG) and tangential magnetostrophy (TM). The calculated CAM variations are examined in reference to two features of the observed length-of-day (LOD) variation, namely, its secular trend and 6year oscillation. We find flow models in either TG or TM state for which the estimated CAM trends agree with the LOD trend. It is necessary for TM models to have their flows dominate at planetary scales, whereas TG models should not be of this scale; otherwise, their CAM trends are too steep. These two distinct types of flow model appear to correspond to the separate regimes of previous numerical dynamos that are thought to be applicable to the Earth's core. The phase of the subdecadal CAM variation is coherently determined from flow models obtained with extensively varying inversion settings. Multiple sources of model ambiguity need to be allowed for in discussing whether these phase estimates properly represent that of Earth's CAM as an origin of the observed 6year LOD oscillation.
We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas.
As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.
We find necessary conditions for a second order ordinary differential equation to be equivalent to the Painleve III equation under a general point transformation. Their sufficiency is established by reduction to known results for the equations of the form y ' = f (x, y). We consider separately the generic case and the case of reducibility to an autonomous equation. The results are illustrated by the primary resonance equation.
Context. The theoretically studied impact of rapid rotation on stellar evolution needs to be compared with these results of high-resolution spectroscopy-velocimetry observations. Early-type stars present a perfect laboratory for these studies. The prototype A0 star Vega has been extensively monitored in recent years in spectropolarimetry. A weak surface magnetic field was detected, implying that there might be a (still undetected) structured surface. First indications of the presence of small amplitude stellar radial velocity variations have been reported recently, but the confirmation and in-depth study with the highly stabilized spectrograph SOPHIE/OHP was required.
Aims. The goal of this article is to present a thorough analysis of the line profile variations and associated estimators in the early-type standard star Vega (A0) in order to reveal potential activity tracers, exoplanet companions, and stellar oscillations.
Methods. Vega was monitored in quasi-continuous high-resolution echelle spectroscopy with the highly stabilized velocimeter SOPHIE/OHP. A total of 2588 high signal-to-noise spectra was obtained during 34.7 h on five nights (2 to 6 of August 2012) in high-resolution mode at R = 75 000 and covering the visible domain from 3895 6270 angstrom. For each reduced spectrum, least square deconvolved equivalent photospheric profiles were calculated with a T-eff = 9500 and log g = 4.0 spectral line mask. Several methods were applied to study the dynamic behaviour of the profile variations (evolution of radial velocity, bisectors, vspan, 2D profiles, amongst others).
Results. We present the discovery of a spotted stellar surface on an A-type standard star (Vega) with very faint spot amplitudes Delta F/Fc similar to 5 x 10(-4). A rotational modulation of spectral lines with a period of rotation P = 0.68 d has clearly been exhibited, unambiguously confirming the results of previous spectropolarimetric studies. Most of these brightness inhomogeneities seem to be located in lower equatorial latitudes. Either a very thin convective layer can be responsible for magnetic field generation at small amplitudes, or a new mechanism has to be invoked to explain the existence of activity tracing starspots. At this stage it is difficult to disentangle a rotational from a stellar pulsational origin for the existing higher frequency periodic variations.
Conclusions. This first strong evidence that standard A-type stars can show surface structures opens a new field of research and ask about a potential link with the recently discovered weak magnetic field discoveries in this category of stars.
A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S-4 or diffeomorphic to CP2.
This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S-2 x S-2 holds in this class of manifolds.
Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.
We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander.