The effect of internal gravity waves on cloud evolution in sub-stellar atmospheres

Context. Sub-stellar objects exhibit photometric variability which is believed to be caused by a number of processes such as magnetically-driven spots or inhomogeneous cloud coverage. Recent sub-stellar models have shown that turbulent flows and waves, including internal gravity waves, may play an important role in cloud evolution.Aims. The aim of this paper is to investigate the effect of internal gravity waves on dust cloud nucleation and dust growth, and whether observations of the resulting cloud structures could be used to recover atmospheric density information.Methods. For a simplified atmosphere in two dimensions, we numerically solve the governing fluid equations to simulate the effect on dust nucleation and mantle growth as a result of the passage of an internal gravity wave. Furthermore, we derive an expression that relates the properties of the wave-induced cloud structures to observable parameters in order to deduce the atmospheric density.Results. Numerical simulations show that the density, pressure and temperature variations caused by gravity waves lead to an increase of dust nucleation by up to a factor 20, and dust mantle growth rate by up to a factor 1:6, compared to their equilibrium values. Through an exploration of the wider sub-stellar parameter space, we show that in absolute terms, the increase in dust nucleation due to internal gravity waves is stronger in cooler (T dwarfs) and TiO2-rich sub-stellar atmospheres. The relative increase however is greater in warm(L dwarf) and TiO2-poor atmospheres due to conditions less suited for efficient nucleation at equilibrium. These variations lead to banded areas in which dust formation is much more pronounced, and lead to banded cloud structures similar to those observed on Earth. Conclusions. Using the proposed method, potential observations of banded clouds could be used to estimate the atmospheric density of sub-stellar objects

Sixty Years of Fractal Projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Comment: Submitted to proceedings of Fractals and Stochastics

Ergodic directions for billiards in a strip with periodically located obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.Comment: The article is complementary to arXiv:1109.458

Emission heights of coronal bright points on Fe XII radiance map

We study the emission heights of the coronal bright points (BPs) above the photosphere in the bipolar magnetic loops that are apparently associated with them. As BPs are seen in projection against the disk their true emission heights are unknown. The correlation of the BP locations on the Fe XII radiance map from EIT with the magnetic field features (in particular neutral lines) was investigated in detail. The coronal magnetic field was determined by an extrapolation of the photospheric field to different altitudes above the disk. It was found that most BPs sit on or near a photospheric neutral line, but that the emission occurs at a height of about 5 Mm. Some BPs, while being seen in projection, still seem to coincide with neutral lines, although their emission takes place at heights of more than 10 Mm. Such coincidences almost disappear for emissions above 20 Mm. We also projected the upper segments of the 3-D magnetic field lines above different heights, respectively, on to the x-y plane. The shape of each BP was compared with the respective field-line segment nearby. This comparison suggests that most coronal BPs are actually located on the top of their associated magnetic loops. Finally, we calculated for each selected BP region the correlation coefficient between the Fe XII intensity enhancement and the horizontal component of the extrapolated magnetic field vector at the same x-y position in planes of different heights, respectively. We found that for almost all the BP regions we studied the correlation coefficient, with increasing height, increases to a maximal value and then decreases again. The height corresponding to this maximum was defined as the correlation height, which for most bright points was found to range below 20 Mm.Comment: 7 pages, 4 figures, 1 tabl

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

Irreversibility in a simple reversible model

This paper studies a parametrized family of familiar generalized baker maps, viewed as simple models of time-reversible evolution. Mapping the unit square onto itself, the maps are partly contracting and partly expanding, but they preserve the global measure of the definition domain. They possess periodic orbits of any period, and all maps of the set have attractors with well defined structure. The explicit construction of the attractors is described and their structure is studied in detail. There is a precise sense in which one can speak about absolute age of a state, regardless of whether the latter is applied to a single point, a set of points, or a distribution function. One can then view the whole trajectory as a set of past, present and future states. This viewpoint is then applied to show that it is impossible to define a priori states with very large "negative age". Such states can be defined only a posteriori. This gives precise sense to irreversibility -- or the "arrow of time" -- in these time-reversible maps, and is suggested as an explanation of the second law of thermodynamics also for some realistic physical systems.Comment: 15 pages, 12 Postscript figure

Analytical study of the effect of recombination on evolution via DNA shuffling

We investigate a multi-locus evolutionary model which is based on the DNA shuffling protocol widely applied in \textit{in vitro} directed evolution. This model incorporates selection, recombination and point mutations. The simplicity of the model allows us to obtain a full analytical treatment of both its dynamical and equilibrium properties, for the case of an infinite population. We also briefly discuss finite population size corrections

The fractal distribution of haloes

We examine the proposal that a model of the large-scale matter distribution consisting of randomly placed haloes with power-law profile, as opposed to a fractal model, can account for the observed power-law galaxy-galaxy correlations. We conclude that such model, which can actually be considered as a degenerate multifractal model, is not realistic but suggests a new picture of multifractal models, namely, as sets of fractal distributions of haloes. We analyse, according to this picture, the properties of the matter distribution produced in cosmological N-body simulations, with affirmative results; namely, haloes of similar mass have a fractal distribution with a given dimension, which grows as the mass diminishes.Comment: 7 pages, 1 figure (3 EPS files), accepted in Europhysics Letter

How large dimension guarantees a given angle?

We study the following two problems: (1) Given $n\ge 2$ and \al, how large Hausdorff dimension can a compact set A\su\Rn have if $A$ does not contain three points that form an angle \al? (2) Given \al and \de, how large Hausdorff dimension can a %compact subset $A$ of a Euclidean space have if $A$ does not contain three points that form an angle in the \de-neighborhood of \al? An interesting phenomenon is that different angles show different behaviour in the above problems. Apart from the clearly special extreme angles 0 and $180^\circ$, the angles $60^\circ,90^\circ$ and $120^\circ$ also play special role in problem (2): the maximal dimension is smaller for these special angles than for the other angles. In problem (1) the angle $90^\circ$ seems to behave differently from other angles

Equilibrium states of the pressure function for products of matrices

Let $\{M_i\}_{i=1}^\ell$ be a non-trivial family of $d\times d$ complex matrices, in the sense that for any $n\in \N$, there exists $i_1... i_n\in \{1,..., \ell\}^n$ such that $M_{i_1}... M_{i_n}\neq {\bf 0}$. Let $P \colon (0,\infty)\to \R$ be the pressure function of $\{M_i\}_{i=1}^\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property.Comment: 12 pages. To appear in DCD