Discrete Analog of the Burgers Equation

We propose the set of coupled ordinary differential equations dn_j/dt=(n_{j-1})^2-(n_j)^2 as a discrete analog of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.Comment: 6 pages, 5 figure

The environmental implications of upper Paleozoic plant-fossil assemblages with mixtures of wetland and drought-tolerant taxa in tropical Pangea

We evaluate the influences of elevation and climate on the spatio-temporal distribution of wetland and dryland biomes during the Pennsylvanian and early Permian in tropical Pangea. The longstanding “upland model” places drought-tolerant vegetation in elevated habitats, where slope and drainage created moisture-limited substrates under a humid climate that simultaneously promoted peat accumulation in contemporaneous lowlands. Upland plants were periodically transported to, and buried in, lowlands. Rare preservation of dryland vegetation thus reflects its general absence in basins, and taphonomic vagaries of long-distance transport. The alternative “climate model” proposes that drought-tolerant plants dominated tropical habitats when climate was seasonally dry, with wetland vegetation reduced to scattered refugia. Environmental changes attending glacial-interglacial cycles caused alternating wetter-drier conditions, and the relative abundance of wetland versus dryland biomes in basinal lowlands thus varied with climatic oscillations. The paucity of drought-tolerant plants reflects a preservational megabias against habitats with seasonal moisture deficits.The environmental signal of “mixed” plant-fossil assemblages, comprising taxa characteristic of both wetland and dryland biomes, may help resolve these debates. We review key Pennsylvanian and lower Permian mixed assemblages from tropical Euramerican Pangea, and interpret their original habitats and climatic contexts based on multidisciplinary lines of evidence, including sedimentology, taphonomy, physiology, and paleoecology. Evaluations also consider patterns of vegetational distribution and taphonomy in modern tropical environments. We suggest that even a cursory view of current tropical plant distribution exposes flaws in the upland model. Where tropical climate is sufficiently humid to support peat swamps, slopes and elevated habitats do not host drought-tolerant vegetation, but are occupied by plants similar to those in lowland settings. This occurs because equable, high precipitation strongly dampens water-table variation across entire landscapes. Furthermore, taphonomic studies indicate that most plant-fossil assemblages record vegetation living near the burial site. Fossil floras thus reflect environmental conditions near their growth site, excluding an upland origin for most occurrences of drought-tolerant taxa. Conversely, the climate model is consistent with modern tropical vegetational distribution and soundly explains late Paleozoic floristic patterns. When Pangean tropical lowlands experienced seasonally dry conditions, plants tolerant of moisture deficits dominated most habitats, whereas wetland vegetation was restricted to wetter sites with greater preservation potential. This occurred because topographic variations are magnified under seasonal precipitation regimes, creating a complex habitat mosaic with wetland patches in a landscape subject to seasonal drought. Accordingly, we propose that a macrofloral assemblage with even rare drought-tolerant taxa indicates seasonality in the broader landscape.At larger spatio-temporal scales, disagreement also persists about whether tectonic uplift or long-term climatic drying was the primary driver of changes in late Paleozoic floristic patterns and areal extent of tropical peat swamps. We argue that tectonic activity alone cannot explain the drastic reduction in peat swamps or coincident changes in dominance-diversity of wetland vegetation. Rates of plant dispersal and evolution far outpace that of mountain building, and peat-forming wetlands persisted in elevated habitats well into the Late Pennsylvanian. Therefore, progressive late Paleozoic aridification was the most probable driver of changing floral patterns and the distribution of wetland and dryland biomes in tropical Pangea

Discussion on 'Tectonic and environmental controls on Palaeozoic fluvial environments: reassessing the impacts of early land plants on sedimentation'. Journal of the Geological Society, https://doi.org/10.1144/jgs2016-063

The first-order importance of tectonic and environmental controls for terrigenous sediment supply has rarely been questioned, but the role of vegetation in the modification of ancient alluvial signatures has been observed since the mid-20th century (Vogt 1941). Studies of sparsely vegetated rivers (Schumm 1968) and alluvial stratigraphic variation (Cotter 1978; Davies & Gibling 2010) led to observations of (1) plant modulation of alluvial signatures and (2) Palaeozoic facies shifts (PFS): unidirectional changes to facies diversity and frequency, in stratigraphic alliance with the plant fossil record. One PFS is the Siluro-Devonian appearance of mud-rich, architecturally complex alluvium, traditionally ascribed to meandering rivers, and sedimentologically distinct from pre-vegetation strata (Davies & Gibling 2010; Long 2011). Using selected secondary data, Santos et al. (2017) dispute the correlation of these observations using three key points, as follows. (1) The mid-Palaeozoic was typified by orogenic assembly of low-gradient equatorial continents and elevated sea-level, which led to tropical weathering (abundant fine sediment) and extensive alluvial plains. This drove the PFS by promoting river meandering independently of vegetation. (2) Meandering does not require vegetation; this is shown by examples in Precambrian deposits, on other planets, and in ‘non-vegetated’ deserts. Meandering rivers were more abundant than the pre-vegetation rock record suggests, owing to selective bypass and deflation of fine material. (3) Early Siluro-Devonian (meaning Ludlow–Early Devonian) land plants were too small, their biomass and cover too limited, and their wetland habitat too narrow to have stabilized meandering channels, influencing landscape little more than earlier microbial communities. We contest the conclusions and method of the paper, and deal with each point in turn

Evaporation and fluid dynamics of a sessile drop of capillary size

Theoretical description and numerical simulation of an evaporating sessile drop are developed. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. A shape of the rotationally symmetric drop is determined within the quasistationary approximation. Nonstationary effects in the diffusion of the vapor are also taken into account. Simulation results agree well with the data of evaporation rate measurements for the toluene drop. Marangoni forces associated with the temperature dependence of the surface tension, generate fluid convection in the sessile drop. Our results demonstrate several dynamical stages of the convection characterized by different number of vortices in the drop. During the early stage the street of vortices arises near a surface of the drop and induces a non-monotonic spatial distribution of the temperature over the drop surface. The initial number of near-surface vortices in the drop is controlled by the Marangoni cell size which is similar to that given by Pearson for flat fluid layers. This number quickly decreases with time, resulting in three bulk vortices in the intermediate stage. The vortices finally transform into the single convection vortex in the drop, existing during about 1/2 of the evaporation time.Comment: 23 pages, 12 figure

Evaporation and explosion of liquid drops on a heated surface

The literature pertinent to various aspects of drop evaporation on a heated surface is reviewed. Both the laser shadowgraphic and direct photographic methods are employed to study thermal stability and flow structures in evaporating drops in all heating regimes. It is revealed that four flow regions exist in stable and unstable type drops at low liquid-film type vaporization regime. As the surface temperature is raised, the flow regions reduce to two. In the nucleate-boiling type vaporization regime, the interfacial flow structure changes due to a reduction in the Marangoni number as well as the dielectric constant of the liquid. An evidence of bubble growth in the drops is disclosed. The micro explosion of drops is found to occur in the transition-boiling type heating range. No drop explosion takes place in the spheriodal vaporization regime except when the drop rolls on to a microscratch on the heating surface. It is concluded that the mechanisms for triggering drop explosion include the spontaneous nucleation and growth phenomena and the destabilization of film boiling.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47061/1/348_2004_Article_BF00266263.pd