The theoretical description of trapped weakly-interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the 'maze' of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This Tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain 'classes of approaches' identified by their physical content, rather than their particular mathematical implementation.This Tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions such as the occupation number representation of second quantization are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean-field, number-conserving and stochastic) according to their particular needs. In addition to its 'training element', we hope that this Tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.

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The distribution of the orbits of close-in exoplanets shows evidence for on-going removal and destruction by tides. Tides raised on a planet's host star cause the planet's orbit to decay, even after the orbital eccentricity has dropped to zero. Comparison of the observed orbital distribution and predictions of tidal theory show good qualitative agreement, suggesting tidal destruction of close-in exoplanets is common. The process can explain the observed cut-off in small a-values, the clustering of orbital periods near three days, and the relative youth of transiting planets. Contrary to previous considerations, a mechanism to stop the inward migration of close-in planets at their current orbits is not necessarily required. Planets nearing tidal destruction may be found with extremely small a, possibly already stripped of any gaseous envelope. The recently discovered CoRoT-Exo-7 b may be an example of such a planet and will probably be destroyed by tides within the next few Gyrs. Also, where one or more planets have already been accreted, a star may exhibit an unusual composition and/or spin rate.

We examine the radius evolution of close-in giant planets with a planet evolution model that couples the orbital-tidal and thermal evolution. For 45 transiting systems, we compute a large grid of cooling/contraction paths forward in time, starting from a large phase space of initial semi-major axes and eccentricities. Given observational constraints at the current time for a given planet (semi-major axis, eccentricity, and system age) we find possible evolutionary paths that match these constraints, and compare the calculated radii to observations. We find that tidal evolution has two effects. First, planets start their evolution at larger semi-major axis, allowing them to contract more efficiently at earlier times. Second, tidal heating can significantly inflate the radius when the orbit is being circularized, but this effect on the radius is short-lived thereafter. Often circularization of the orbit is proceeded by a long period while the semi-major axis slowly decreases. Some systems with previously unexplained large radii that we can reproduce with our coupled model are HAT-P-7, HAT-P-9, WASP-10, and XO-4. This increases the number of planets for which we can match the radius from 24 (of 45) to as many as 35 for our standard case, but for some of these systems we are required to be viewing them at a special time around the era of current radius inflation. This is a concern for the viability of tidal inflation as a general mechanism to explain most inflated radii. Also, large initial eccentricities would have to be common. We also investigate the evolution of models that have a floor on the eccentricity, as may be due to a perturber. In this scenario we match the extremely large radius of WASP-12b. This work may cast some doubt on our ability to accurately determine the interior heavy element enrichment of normal, non-inflated close-in planets, because of our dearth of knowledge about these planet's previous orbital-tidal histories. Finally, we find that the end state of most close-in planetary systems is disruption of the planet as it moves ever closer to its parent star.

Extrasolar planets close to their host stars have likely undergone significant tidal evolution since the time of their formation. Tides probably dominated their orbital evolution once the dust and gas cleared away, and as the orbits evolved there was substantial tidal heating within the planets. The tidal heating history of each planet may have contributed significantly to the thermal budget governing the planet's physical properties, including its radius, which in many cases may be measured by observing transit events. Typically, tidal heating increases as a planet moves inward toward its star and then decreases as its orbit circularizes. Here we compute the plausible heating histories for several planets with measured radii, using the same tidal parameters for the star and planet that have been shown to reconcile the eccentricity distribution of close-in planets with other extrasolar planets. Several planets are discussed, including, for example, HD 209458b, which may have undergone substantial tidal heating during the past billion years, perhaps enough to explain its large measured radius. Our models also show that GJ 876d may have experienced tremendous heating and is probably not a solid, rocky planet. Theoretical models should include the role of tidal heating, which is large but time-varying.

This study examines mesoscale convective systems (MCSs) over western equatorial Africa using data from the Tropical Rainfall Measuring Mission (TRMM) satellite. This region experiences some of the world’s most intense thunderstorms and highest lightning frequency, but has low rainfall relative to other equatorial regions. The analyses of MCS activity include the frequency of occurrence, diurnal and annual cycles, and associated volumetric and convective rainfall. Also evaluated is the lightning activity associated with the MCSs. Emphasis is placed on the diurnal cycle and on the continental-scale motion fields in this region. The diurnal cycle shows a maximum in MCS count around 1500–1800 LT, a morning minimum, and substantial activity during the night; there is little seasonal variation in the diurnal cycle, suggesting stationary influences such as orography. Our analysis shows four maxima in MCS activity, three of which are related to local geography (two orographic and one over Lake Victoria). The fourth coincides with a midtropospheric convergence maximum in the right entrance quadrant of the African easterly jet of the Southern Hemisphere (AEJ-S). This maximum is substantially stronger in the September–November rainy season, when the jet is well developed, than in the March–May rainy season, when the jet is absent. Lightning frequency and flashes per MCS are also greatest during September–November; maxima occur in the right entrance quadrant of the AEJ-S. The lightning maximum is somewhat south of the MCS maximum and coincides with the low-lying areas of central Africa. Overall, the results of this study suggest that large-scale topography plays a critical role in the spatial and diurnal patterns of convection, lightning, and rainfall in this region. More speculative is the role of the AEJ-S, but this preliminary analysis suggests that it does play a role in the anomalous intensity of convection in western equatorial Africa.

We solve the time-dependent Gross-Pitaevskii in 2-D to simulate the flow of an object through a dilute Bose condensate trapped in a harmonic well. We demonstrate vortex formation and analyze the process in terms of the accumulation of phase-slip and the evolution of the fluid velocity.

A two-dimensional rapidly rotating Bose-Einstein condensate in an anharmonic trap with quadratic and quartic radial confinement is studied analytically with the Thomas-Fermi approximation and numerically with the full time-independent Gross-Pitaevskii equation. The quartic trap potential allows the rotation speed Ω to exceed the radial harmonic frequency ω ⊥ . In the regime Ω ω ⊥ , the condensate contains a dense vortex array (approximated as solid-body rotation for the analytical studies). At a critical angular velocity Ω h , a central hole appears in the condensate. Numerical studies confirm the predicted value of Ω h , even for interaction parameters that are not in the Thomas-Fermi limit. The behavior is also investigated at larger angular velocities, where the system is expected to undergo a transition to a giant vortex (with pure irrotational flow).