Bifurcation study for a vertical channel with constant flux and large aspect ratio

Using suitable coupled Navier-Stokes Equations for an incompressible Newtonian fluid we investigate the linear and non-linear steady state solutions for both a homogeneously and a laterally heated fluid with finite Prandtl Number (Pr=7) in the vertical orientation of the channel. Both models are studied within the Large Aspect Ratio narrow-gap and under constant flux conditions with the channel closed. We use direct numerics to identify the linear stability criterion in parametric terms as a function of Grashof Number (Gr) and streamwise infinitesimal perturbation wavenumber (making use of the generalised Squire’s Theorem). We find higher harmonic solutions at lower wavenumbers with a resonance of 1:3exist, for both of the heating models considered. We proceed to identify 2D secondary steady state solutions, which bifurcate from the laminar state. Our studies show that 2D solutions are found not to exist in certain regions of the pure manifold, where we find that 1:3 resonant mode 2D solutions exist, for low wavenumber perturbations. For the homogeneously heated fluid, we notice a jump phenomenon existing between the pure and resonant mode secondary solutions for very specific wavenumbers .We attempt to verify whether mixed mode solutions are present for this model by considering the laterally heated model with the same geometry. We find mixed mode solutions for the laterally heated model showing that a bridge exists between the pure and 1:3 resonant mode 2D solutions, of which some are stationary and some travelling. Further, we show that for the homogeneously heated fluid that the 2D solutions bifurcate in hopf bifurcations and there exists a manifold where the 2D solutions are stable to Eckhaus criterion, within this manifold we proceed to identify 3D tertiary solutions and find that the stability for said 3D bifurcations is not phase locked to the 2D state. For the homogeneously heated model we identify a closed loop within the neutral stability curve for higher perturbation wavenumubers and analyse the nature of the multiple 2D bifurcations around this loop for identical wavenumber and find that a temperature inversion occurs within this loop. We conclude that for a homogeneously heated fluid it is possible to have abrup ttransitions between the pure and resonant 2D solutions, and that for the laterally heated model there exist a transient bifurcation via mixed mode solutions.

Seasonal growth of the crustose lichen Rhizocarpon geographicum (L.) DC. in South Gwynedd, Wales

Seasonal growth was studied in the slow-growing crustose lichen Rhizocarpon geographicum (L.) DC. in an area of South Gwynedd, Wales. Radial growth rate (RGR) of a sample of 20 thalli was measured in situ at three-month intervals over 51 months on a southeast-facing rock surface. There were five periods of significant growth: July-September of 1993, 1994 and 1995, in January-March of 1996, and in April-June of 1997. In four of these periods, growth coincided with a mean temperature maximum (Tmax) over a three-month period exceeding 15°C and three of the maxima with greater than 450 sunshine hours. Two of the growth maxima coincided with periods of total rainfall exceeding 300 mm and one with greater than 50 rain days in a three-month period. There were no significant linear correlations between RGR and the climatic variables measured. However, there were significant non-linear relationships between RGR and Tmax, the mean temperature minimum (Tmin), the total number of air and ground frosts and the number of rain days in a growth period, the relationship with Tmax being the most significant. Hence, in south Gwynedd, maximum growth of R. geographicum occurs in any season although the period July-September appears to be the most favourable. Relationships between growth and climatic variables were non-linear, temperature having the most significant influence on seasonal growth. ©2006 Balaban.

Variation in hypothallus width and the growth of the lichen Rhizocarpon geographicum (L.) DC.

Variations in hypothallus width were studied in relation to radial growth in the lichen Rhizocarpon geographicum (L.) DC. in South Gwynedd, Wales, UK. Variations were present both within and between thalli and in successive three-month growth periods, but there was no significant variation associated with thallus size. In individual thalli, there were increases and reductions in hypothallus width in successive three-month growth periods attributable to hypothallus growth and changes at the margin of the areolae. Total radial growth over 18 months was positively correlated with initial hypothallus width. These results suggest: 1) individual thalli of similar size vary considerably in hypothallus width, 2) fluctuations in the location of the margin of the areolae in successive three month periods is an important factor determining this variability, 3) hypothallus width predicts subsequent radial growth over 18 months, and 4) variation in hypothallus; width is a factor determining between thallus variability in radial growth rates in yellow-green species of Rhizocarpon.

Soredial dispersal from individual soralia in the lichen Hypogymnia physodes (L.) Nyl.

Soredial dispersal from individual soralia of Hypogymnia physodes (L.) Nyl. was studied in the field under natural conditions and by exposing the soralia to an electric fan. Individual soralia were placed on the adhesive surface of dust particle collectors which were pinned to vertical boards in the field. The majority of soredia that were deposited on the adhesive strips during the experiments were found within 1 cm of the source soralium. Deposition was studied over 6 successive days under natural conditions. Significantly fewer soredia were deposited from soralia after removal of mature accumulations and from soralia taken from moist thalli compared with soralia from air dry thalli. In addition, there was a decline in soredial deposition over the 6 days. The influence of wind speed and initial thallus moisture content on soredial deposition over short intervals of time was studied using an electric fan. More soredia and larger soredial clusters were deposited from air dry than moist soralia at all wind speeds. Variation in wind speed between 4 and 9 m/sec had little effect on soredial deposition. Deposition of soredia was also studied using the fan over successive 5-min intervals. Large numbers of soredia were deposited during the first 5-min period. Deposition then declined but recovered after about four 5-min periods. In all experiments there were differences between individual soralia in total numbers of soredia deposited and in the pattern of deposition over time. These results suggest (1) soredia accumulate on soralia and these deposits may be gradually or rapidly depleted in the field, (2) that after the release of soredial accumulations some newly exposed soredia may be rapidly dispersed, (3) a high initial thallus moisture content inhibits soredial release and (4) variation in wind speed is less important than moisture in influencing soredial deposition. The results may help to explain the intermittent pattern of soredial deposition and the poor correlations between deposition and climatic factors observed previously in the field. © 1992.

Experimental studies on the distribution of the sexes of Mercurialis perennis L.

1) In a beechwood, male shoots of Mercurialis perennis were more frequent than female shoots in canopy openings, while females were more frequent at treebases than in canopy openings. 2) Flowering shoots were more frequent than non-flowering in openings. 3) The pH, moisture and organic content of the soil were similar at treebases and openings. The light climate was similar in both habitats in April but greater in the openings in August. 4) Removal of canopies of Pteridium aquilinum and Rubus fruticosus, above populations of M. perennis, resulted in a greater increase in the numbers of male shoots than of female shoots. 5) These results suggest that male and female plants differ in numbers, or growth, in different woodland microhabitats and that these differences are correlated with the above-ground environment (e.g. light).

Dispersal of soredia from individual soralia of the lichen Hypogymnia physodes (L.) Nyl. in a simple wind tunnel

Dispersal of soredia from individual soralia of the lichen Hypogymnia physodes (L.) Nyl. was studied using a simple wind tunnel constructed in the field. Individual lobes with terminal soralia were placed in the wind tunnel on the adhesive surface of dust particle collectors. Air currents produced by a fan were directed over the surface of the lobes. The majority of soredia were deposited within 5 cm of the source soralium but some soredia were dispersed to at least 80 cm at a wind speed of 6 m s-1. Variation in wind speed had no statistically significant effect on the total number of soredial clusters deposited averaged over soralia but the mean size of cluster and the distance dispersed were greater at higher wind speeds. The number of soredia deposited was dependent on the orientation of the soralium to the air currents. More soredia were deposited with the soralium facing the fan at a wind speed of 9 m s-1. Moisture in the form of a fine mist reduced substantially the number of soredia deposited at a wind speed of 6 m s-1 but had no effect on the mean number of soredia per cluster or on the mean distance dispersed. The data suggest: (1) that wind dispersal from an individual soralium is influenced by wind speed, the location of the soralium on the thallus and the level of moisture and (2) that air currents directed over the surfaces of thalli located on the upper branches of trees would effectively disperse soredia of H. physodes vertically and horizontally within a tree canopy. © 1994.

Dispersal, establishment and survival of soredia and fragments of the lichen, Hypogymnia physodes (L.) Nyl.

Dispersal, establishment and survival were studied in a population of Hypogymnia physodes (L.) Nyl. growing on an isolated tree (Prunus blireiana L.) at a site in North Seattle, U.S.A. Lichen propagules were trapped daily on adhesive strips pinned to various sites on the tree over a period of 36 days. Both soredia and fragments were deposited on the strips, particularly on the upper branches of the tree, with soredia being considerably more numerous than fragments. Daily variation in total soredia deposited did not correlate with 10 climatic variables including wind speed, relative humidity and average temperature. Establishment and survival of propagules were studied by introducing soredia and fragments into various sites on the bark and by observing the distribution of small thalli in different microsites. Microtopography of bark significantly influenced establishment and survival. Survival was poor on smooth bark compared with survival on algal or lichen crusts and on rough bark. Survival of soredia did not vary significantly at different locations on the tree. It is likely that colonization of the tree by H. physodes occurs largely by soredia. Colonization appears to be limited more by the range of dispersal over the tree than by differential survival over different parts of the tree.

Carbohydrates in the hypothallus and areolae of the crustose lichen Rhizocarpon geographicum (L.) DC

Carbohydrate concentrations in the marginal hypothallus and areolae of the crustose lichen Rhizocarpon geographicum (L.) DC. were measured in north Wales, U.K. using gas chromatography. Ribitol, arabitol, and mannitol were the most abundant carbohydrates while a- glucose ß-glucose, fructose, sucrose, and trehalose were present in smaller amounts. The concentrations of arabitol, ribitol, mannitol, fructose, and a-glucose were greater in the areolae while the concentration of trehalose was greater in the hypothallus. Concentrations of carbohydrates varied little between sample days. Concentrations of polyols in the hypothallus were not correlated with those in the areolae. These results suggest: 1) the hypothallus has a lower demand for carbohydrates than the areolae or there is limited transport from areolae to hypothallus, 2) increased trehalose in the non-lichenised hypothallus may be an adaptation to withstand stress and desiccation, and 3) polyols are partitioned differently in the hypothallus and areolae.