Nonlinear saturation of baroclinic instability: part III: bounds on the energy

Rigorous upper bounds are derived on the saturation amplitude of baroclinic instability in the two-layer model. The bounds apply to the eddy energy and are obtained by appealing to a finite amplitude conservation law for the disturbance pseudoenergy. These bounds are to be distinguished from those derived in Part I of this study, which employed a pseudomomentum conservation law and provided bounds on the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity. Bounds on the eddy energy are worked out for a general class of unstable westerly jets. In the special case of the Phillips model of baroclinic instability, and in the limit of infinitesimal initial eddy amplitude, the bound states that the eddy energy cannot exceed ϵβ2/6F where ϵ = (U − Ucrit)/Ucrit is the relative supercriticality. This bound captures the essential dynamical scalings (i.e., the dependence on ϵ, β, and F) of the saturation amplitudes predicted by weakly nonlinear theory, as well as exhibiting remarkable quantitative agreement with those predictions, and is also consistent with heuristic baroclinic adjustment estimates.

Nonlinear saturation of baroclinic instability: part II: continuously stratified fluid

Rigorous upper bounds are derived that limit the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow in a continuously stratified, quasi-geostrophic, semi-infinite fluid. Bounds are obtained bath on the depth-integrated eddy potential enstrophy and on the eddy available potential energy (APE) at the ground. The method used to derive the bounds is essentially analogous to that used in Part I of this study for the two-layer model: it relies on the existence of a nonlinear Liapunov (normed) stability theorem, which is a finite-amplitude generalization of the Charney-Stern theorem. As in Part I, the bounds are valid both for conservative (unforced, inviscid) flow, as well as for forced-dissipative flow when the dissipation is proportional to the potential vorticity in the interior, and to the potential temperature at the ground. The character of the results depends on the dimensionless external parameter γ = f02ξ/β0N2H, where ξ is the maximum vertical shear of the zonal wind, H is the density scale height, and the other symbols have their usual meaning. When γ ≫ 1, corresponding to “deep” unstable modes (vertical scale ≈H), the bound on the eddy potential enstrophy is just the total potential enstrophy in the system; but when γ≪1, corresponding to ‘shallow’ unstable modes (vertical scale ≈γH), the eddy potential enstrophy can be bounded well below the total amount available in the system. In neither case can the bound on the eddy APE prevent a complete neutralization of the surface temperature gradient which is in accord with numerical experience. For the special case of the Charney model of baroclinic instability, and in the limit of infinitesimal initial eddy disturbance amplitude, the bound states that the dimensionless eddy potential enstrophy cannot exceed (γ + 1)2/24&gamma2h when γ ≥ 1, or 1/6;&gammah when γ ≤ 1; here h = HN/f0L is the dimensionless scale height and L is the width of the channel. These bounds are very similar to (though of course generally larger than) ad hoc estimates based on baroclinic-adjustment arguments. The possibility of using these kinds of bounds for eddy-amplitude closure in a transient-eddy parameterization scheme is also discussed.

Nonlinear saturation of baroclinic instability: part I: the two-layer model

A rigorous bound is derived which limits the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow within the context of the two-layer model. The bound is valid for conservative (unforced) flow, as well as for forced-dissipative flow that when the dissipation is proportional to the potential vorticity. The method used to derive the bound relies on the existence of a nonlinear Liapunov (normed) stability theorem for subcritical flows, which is a finite-amplitude generalization of the Charney-Stern theorem. For the special case of the Philips model of baroclinic instability, and in the limit of infinitesimal initial nonzonal disturbance amplitude, an improved form of the bound is possible which states that the potential enstrophy of the nonzonal flow cannot exceed ϵβ2, where ϵ = (U − Ucrit)/Ucrit is the (relative) supereriticality. This upper bound turns out to be extremely similar to the maximum predicted by the weakly nonlinear theory. For unforced flow with ϵ < 1, the bound demonstrates that the nonzonal flow cannot contain all of the potential enstrophy in the system; hence in this range of initial supercriticality the total flow must remain, in a certain sense, “close” to a zonal state.

The energy-cycle analysis of the interactions between shallow and deep atmospheric convection

Interactions between different convection modes can be investigated using an energy–cycle description under a framework of mass–flux parameterization. The present paper systematically investigates this system by taking a limit of two modes: shallow and deep convection. Shallow convection destabilizes itself as well as the other convective modes by moistening and cooling the environment, whereas deep convection stabilizes itself as well as the other modes by drying and warming the environment. As a result, shallow convection leads to a runaway growth process in its stand–alone mode, whereas deep convection simply damps out. Interaction between these two convective modes becomes a rich problem, even when it is limited to the case with no large–scale forcing, because of these opposing tendencies. Only if the two modes are coupled at a proper level can a self–sustaining system arise, exhibiting a periodic cycle. The present study establishes the conditions for self–sustaining periodic solutions. It carefully documents the behaviour of the two mode system in order to facilitate the interpretation of global model behaviours when this energy–cycle is implemented as a closure into a convection parameterization in future.

Ranging characteristics of the domestic cat (Felis catus) in an urban environment

In many countries, high densities of domestic cats (Felis catus) are found in urban habitats where they have the potential to exert considerable predation pressure on their prey. However, little is known of the ranging behaviour of cats in the UK. Twenty cats in suburban Reading, UK, were fitted with GPS trackers to quantify movement patterns. Cats were monitored during the summer and winter for an average of 6.8 24 h periods per season. Mean daily area ranged (95 % MCP) was 1.94 ha. Including all fixes, mean maximum area ranged was 6.88 ha. These are broadly comparable to those observed in urban areas in other countries. Daily area ranged was not affected by the cat’s sex or the season, but was significantly larger at night than during the day. There was no relationship between area ranged and habitat availability. Taking available habitat into account, cat ranging area contained significantly more garden and other green space than urban habitats. If cats were shown to be negatively affecting prey populations, one mitigation option for consideration in housing developments proposed near important wildlife sites would be to incorporate a ‘buffer zone’ in which cat ownership was not permitted. Absolute maximum daily area ranged by a cat in this study was 33.78 ha. This would correspond to an exclusory limit of approximately 300–400 m to minimise the negative effects of cat predation, but this may need to be larger if cat ranging behaviour is negatively affected by population density

Analytical and numerical solutions describing the inward solidification of a binary melt

We present a mathematical model describing the inward solidification of a slab, a circular cylinder and a sphere of binary melt kept below its equilibrium freezing temperature. The thermal and physical properties of the melt and solid are assumed to be identical. An asymptotic method, valid in the limit of large Stefan number is used to decompose the moving boundary problem for a pure substance into a hierarchy of fixed-domain diffusion problems. Approximate, analytical solutions are derived for the inward solidification of a slab and a sphere of a binary melt which are compared with numerical solutions of the unapproximated system. The solutions are found to agree within the appropriate asymptotic regime of large Stefan number and small time. Numerical solutions are used to demonstrate the dependence of the solidification process upon the level of impurity and other parameters. We conclude with a discussion of the solutions obtained, their stability and possible extensions and refinements of our study.

Evaluation of coupled ocean-atmosphere simulations of northern hemisphere extratropical climates in the mid-Holocene

We have used the BIOME4 biogeography–biochemistry model and comparison with palaeovegetation data to evaluate the response of six ocean–atmosphere general circulation models to mid-Holocene changes in orbital forcing in the mid- to high-latitudes of the northern hemisphere. All the models produce: (a) a northward shift of the northern limit of boreal forest, in response to simulated summer warming in high-latitudes. The northward shift is markedly asymmetric, with larger shifts in Eurasia than in North America; (b) an expansion of xerophytic vegetation in mid-continental North America and Eurasia, in response to increased temperatures during the growing season; (c) a northward expansion of temperate forests in eastern North America, in response to simulated winter warming. The northward shift of the northern limit of boreal forest and the northward expansion of temperate forests in North America are supported by palaeovegetation data. The expansion of xerophytic vegetation in mid-continental North America is consistent with palaeodata, although the extent may be over-estimated. The simulated expansion of xerophytic vegetation in Eurasia is not supported by the data. Analysis of an asynchronous coupling of one model to an equilibrium-vegetation model suggests vegetation feedback exacerbates this mid-continental drying and produces conditions more unlike the observations. Not all features of the simulations are robust: some models produce winter warming over Europe while others produce winter cooling. As a result, some models show a northward shift of temperate forests (consistent with, though less marked than, the expansion shown by data) and others produce a reduction in temperate forests. Elucidation of the cause of such differences is a focus of the current phase of the Palaeoclimate Modelling Intercomparison Project.

Palaeovegetation of China: a pollen data-based synthesis for the mid-Holocene and last glacial maximum.

Pollen data from China for 6000 and 18,000 14C yr bp were compiled and used to reconstruct palaeovegetation patterns, using complete taxon lists where possible and a biomization procedure that entailed the assignment of 645 pollen taxa to plant functional types. A set of 658 modern pollen samples spanning all biomes and regions provided a comprehensive test for this procedure and showed convincing agreement between reconstructed biomes and present natural vegetation types, both geographically and in terms of the elevation gradients in mountain regions of north-eastern and south-western China. The 6000 14C yr bp map confirms earlier studies in showing that the forest biomes in eastern China were systematically shifted northwards and extended westwards during the mid-Holocene. Tropical rain forest occurred on mainland China at sites characterized today by either tropical seasonal or broadleaved evergreen/warm mixed forest. Broadleaved evergreen/warm mixed forest occurred further north than today, and at higher elevation sites within the modern latitudinal range of this biome. The northern limit of temperate deciduous forest was shifted c. 800 km north relative to today. The 18,000 14C yr bp map shows that steppe and even desert vegetation extended to the modern coast of eastern China at the last glacial maximum, replacing today’s temperate deciduous forest. Tropical forests were excluded from China and broadleaved evergreen/warm mixed forest had retreated to tropical latitudes, while taiga extended southwards to c. 43°N.

The response of vegetation on the Andean flank in western Amazonia to Pleistocene climate change

A reconstruction of past environmental change from Ecuador reveals the response of lower montane forest on the Andean flank in western Amazonia to glacial-interglacial global climate change. Radiometric dating of volcanic ash indicates that deposition occurred ~324,000 to 193,000 years ago during parts of Marine Isotope Stages 9, 7, and 6. Fossil pollen and wood preserved within organic sediments suggest that the composition of the forest altered radically in response to glacial-interglacial climate change. The presence of Podocarpus macrofossils ~1000 meters below the lower limit of their modern distribution indicates a relative cooling of at least 5°C during glacials and persistence of wet conditions. Interglacial deposits contain thermophilic palms suggesting warm and wet climates. Hence, global temperature change can radically alter vegetation communities and biodiversity in this region.

Quantifying the role of biosphere-atmosphere feedbacks in climate change: coupled model simulation for 6000 years B.P. and comparison with palaeodata for northern Eurasia and northern Africa

The LMD AGCM was iteratively coupled to the global BIOME1 model in order to explore the role of vegetation-climate interactions in response to mid-Holocene (6000 y BP) orbital forcing. The sea-surface temperature and sea-ice distribution used were present-day and CO2 concentration was pre-industrial. The land surface was initially prescribed with present-day vegetation. Initial climate “anomalies” (differences between AGCM results for 6000 y BP and control) were used to drive BIOME1; the simulated vegetation was provided to a further AGCM run, and so on. Results after five iterations were compared to the initial results in order to identify vegetation feedbacks. These were centred on regions showing strong initial responses. The orbitally induced high-latitude summer warming, and the intensification and extension of Northern Hemisphere tropical monsoons, were both amplified by vegetation feedbacks. Vegetation feedbacks were smaller than the initial orbital effects for most regions and seasons, but in West Africa the summer precipitation increase more than doubled in response to changes in vegetation. In the last iteration, global tundra area was reduced by 25% and the southern limit of the Sahara desert was shifted 2.5 °N north (to 18 °N) relative to today. These results were compared with 6000 y BP observational data recording forest-tundra boundary changes in northern Eurasia and savana-desert boundary changes in northern Africa. Although the inclusion of vegetation feedbacks improved the qualitative agreement between the model results and the data, the simulated changes were still insufficient, perhaps due to the lack of ocean-surface feedbacks.

The interaction of moist convection and mid-level dry air in the advance of the onset of the Indian monsoon

The advance of the onset of the Indian monsoon is here explained in terms of a balance between the low-level monsoon flow and an over-running intrusion of mid-tropospheric dry air. The monsoon advances, over a period of about 6 weeks, from the south of the country to the northwest. Given that the low-level monsoon winds are westerly or southwesterly, and the midlevel winds northwesterly, the monsoon onset propagates upwind relative to midlevel flow, and perpendicular to the low-level flow, and is not directly caused by moisture flux toward the northwest. Lacking a conceptual model for the advance means that it has been hard to understand and correct known biases in weather and climate prediction models. The mid-level northwesterlies form a wedge of dry air that is deep in the far northwest of India and over-runs the monsoon flow. The dry layer is moistened from below by shallow cumulus and congestus clouds, so that the profile becomes much closer to moist adiabatic, and the dry layer is much shallower in the vertical, toward the southeast of India. The profiles associated with this dry air show how the most favourable environment for deep convection occurs in the south, and onset occurs here first. As the onset advances across India, the advection of moisture from the Arabian Sea becomes stronger, and the mid-level dry air is increasingly moistened from below. This increased moistening makes the wedge of dry air shallower throughout its horizontal extent, and forces the northern limit of moist convection to move toward the northwest. Wetting of the land surface by rainfall will further reinforce the north-westward progression, by sustaining the supply of boundary layer moisture and shallow cumulus. The local advance of the monsoon onset is coincident with weakening of the mid-level northwesterlies, and therefore weakened mid-level dry advection.

The atomic structure of low-index surfaces of the intermetallic compound InPd

The intermetallic compound InPd (CsCl type of crystal structure with a broad compositional range) is considered as a candidate catalyst for the steam reforming of methanol. Single crystals of this phase have been grown to study the structure of its three low-index surfaces under ultra-high vacuum conditions, using low energy electron diffraction (LEED), X-ray photoemission spectroscopy (XPS), and scanning tunneling microscopy (STM). During surface preparation, preferential sputtering leads to a depletion of In within the top few layers for all three surfaces. The near-surface regions remain slightly Pd-rich until annealing to ∼580 K. A transition occurs between 580 and 660 K where In segregates towards the surface and the near-surface regions become slightly In-rich above ∼660 K. This transition is accompanied by a sharpening of LEED patterns and formation of flat step-terrace morphology, as observed by STM. Several superstructures have been identified for the different surfaces associated with this process. Annealing to higher temperatures (≥750 K) leads to faceting via thermal etching as shown for the (110) surface, with a bulk In composition close to the In-rich limit of the existence domain of the cubic phase. The Pd-rich InPd(111) is found to be consistent with a Pd-terminated bulk truncation model as shown by dynamical LEED analysis while, after annealing at higher temperature, the In-rich InPd(111) is consistent with an In-terminated bulk truncation, in agreement with density functional theory (DFT) calculations of the relative surface energies. More complex surface structures are observed for the (100) surface. Additionally, individual grains of a polycrystalline sample are characterized by micro-spot XPS and LEED as well as low-energy electron microscopy. Results from both individual grains and “global” measurements are interpreted based on comparison to our single crystals findings, DFT calculations and previous literature.

Large deviations of Rouse polymer chain: first passage problem

The purpose of this paper is to investigate several analytical methods of solving first passage (FP) problem for the Rouse model, a simplest model of a polymer chain. We show that this problem has to be treated as a multi-dimensional Kramers' problem, which presents rich and unexpected behavior. We first perform direct and forward-flux sampling (FFS) simulations, and measure the mean first-passage time $\tau(z)$ for the free end to reach a certain distance $z$ away from the origin. The results show that the mean FP time is getting faster if the Rouse chain is represented by more beads. Two scaling regimes of $\tau(z)$ are observed, with transition between them varying as a function of chain length. We use these simulations results to test two theoretical approaches. One is a well known asymptotic theory valid in the limit of zero temperature. We show that this limit corresponds to fully extended chain when each chain segment is stretched, which is not particularly realistic. A new theory based on the well known Freidlin-Wentzell theory is proposed, where dynamics is projected onto the minimal action path. The new theory predicts both scaling regimes correctly, but fails to get the correct numerical prefactor in the first regime. Combining our theory with the FFS simulations lead us to a simple analytical expression valid for all extensions and chain lengths. One of the applications of polymer FP problem occurs in the context of branched polymer rheology. In this paper, we consider the arm-retraction mechanism in the tube model, which maps exactly on the model we have solved. The results are compared to the Milner-McLeish theory without constraint release, which is found to overestimate FP time by a factor of 10 or more.

Thermoelectric properties of TiS2 mechanically alloyed compounds

Bulk polycrystalline samples in the series Ti1−xNbxS2 (0 ≤ x ≤ 0.075) were prepared using mechanical alloying synthesis and spark plasma sintering. X-ray diffraction analysis coupled with high resolution transmission electron microscopy indicates the formation of trigonal TiS2 by high energy ball-milling. The as-synthesized particles consist of pseudo-ordered TiS2 domains of around 20–50 nm, joined by bent atomic planes. This bottom-up approach leads, after spark plasma sintering, to homogeneous solid solutions, with a niobium solubility limit of x = 0.075. Microstructural observations evidence the formation of small crystallites in the bulk compounds with a high density of stacking faults. The large grain boundary concentration coupled with the presence of planar defects, leads to a substantial decrease in the thermal conductivity to 1.8 W/mK at 700 K. This enables the figure of merit to reach ZT = 0.3 at 700 K for x = 0.05, despite the lower electron mobility in mechanically alloyed samples due to small crystallite/grain size and structural defects.

What is the shape of an air bubble on a liquid surface?

We have calculated the equilibrium shape of the axially symmetric meniscus along which a spherical bubble contacts a flat liquid surface, by analytically integrating the Young-Laplace equation in the presence of gravity, in the limit of large Bond numbers. This method has the advantage that it provides semi-analytical expressions for key geometrical properties of the bubble in terms of the Bond number. Results are in good overall agreement with experimental data and are consistent with fully numerical (Surface Evolver) calculations. In particular, we are able to describe how the bubble shape changes from hemispherical, with a shallow flat bottom, to lenticular, with a deeper, curved bottom, as the Bond number is decreased.