Singular solutions of a modified two-component Camassa-Holm equation

The Camassa-Holm equation (CH) is a well known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow dependence on average density as well as pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially-confined initial data. Numerical results for MCH2 are given and compared with the pure CH2 case. These numerics show that the modification in MCH2 to introduce average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for MCH2 shows a new asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.Comment: 25 pages, 11 figure

Parinaud's syndrome – A rare presentation of clinically isolated syndrome

We present a 26 year old Pakistani lady with first presentation of a demyelinating event, presenting as Parinaud's syndrome. The video demonstrates a convergence–retraction nystagmus on upgaze and failure of accommodation, and her brain imaging confirms a corresponding pre-tectal contrast enhancing T2 hyperintense lesion suggestive of demyelination. A review of the literature is presented

Carbohydrate utilization in obese subjects after an oral load of 100 g naturally-labelled [13C] glucose

1. Total carbohydrate (CHO) and ingested glucose oxidation was measured in five obese subjects with normal glucose tolerance after an oral load of 100g naturally-labelled [13C]glucose using indirect calorimetry and mass spectrometry respectively. 2. CHO utilization rate (107 ± 14 mg/min in the post-absorptive state) increased 30 min after the glucose load to reach a plateau (245±25 mg/min) between 90 and 120 min. It then decreased to basal values at 330 min. Cumulative CHO oxidation over 480 min was 66±7 g and the CHO oxidized above basal levels was 26 ± 7g. 3. Enrichment of expired carbon dioxide with 13c began at 45 min and maximum values were observed between 210 and 300 min. At 480 min, cumulative oxidation of the ingested glucose was 24± 2 g. 4. Compared with controls, the obese subjects exhibit an impairment of CHO utilization which precedes glucose intolerance. This impairment can be explained by an increased availability of free fatty acids which favours lipid oxidation at the expense of ingested [13C]glucose oxidatio

Thermic effect of glucose in obese subjects studied by direct and indirect calorimetry

1. The thermic effect of a glucose load (50 g) was studied in ten control and eleven obese female subjects, using both direct and indirect calorimetry simultaneously. Experiments were done under conditions of thermal equilibrium (28° and 30% relative humidity).. 2. Thermal balance (heat production measured by indirect calorimetry minus heat losses measured directly) was negative in the control group during the fasting period (heat deficit - 14.2 ± 5.0 kJ/m2 per h), whereas that of the obese group was in equilibrium (+ 1.4 ± 4.8 kJ/m2 per h).. 3. After the glucose load, metabolic rate increased 13.0 ± 1.5 and 5.2 ± 1.3% in the control and obese groups respectively.. 4. In contrast to the metabolic rate, total heat losses were not significantly altered in either group after the glucose load. Total heat losses of the obese group were significantly lower than those of the control group throughout the experimental period.. 5. During the experiments the amount of heat stored was increased in both groups. Thermal balance in the control group became positive while that of the obese group remained positive.. 6. During the fasting period, the control subjects oxidized more carbohydrates (90.4 mg/min) than lipids (68.8 mg/min), whereas obese subjects oxidized more lipids (103.7 mg/min) than carbohydrates (50.2 mg/min). After the glucose load, the oxidation rate of carbohydrates was increased in both groups to 158.1 mg/min in control subjects and 95.6 mg/min in obese subjects.. 7. The mean skin temperature of the control subjects was significantly higher than that of the obese subjects and remained higher throughout the postprandial period.. 8. These results indicate that: (a) during the fasting period, the energy sources utilized and the thermal balance of the two groups were different; (b) the thermic effect of glucose was less in the obese subjects and, therefore, might be a factor contributing to their low energy expenditur

Stratified gravity currents in porous media

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model

Capillary-gravity waves: The effect of viscosity on the wave resistance

The effect of viscosity on the wave resistance experienced by a 2d perturbation moving at uniform velocity over the free surface of a fluid is investigated. The analysis is based on Rayleigh's linearized theory of capillary-gravity waves. It is shown in particular that the wave resistance remains bounded as the velocity of the perturbation approches the minimun phase speed, unlike what is predicted by the inviscid theory.Comment: Europhysics Letters, in pres

Comparison of carbohydrate utilization in man using indirect calorimetry and mass spectrometry after an oral load of 100 g naturally-labelled [13C]glucose

1. Carbohydrate (CHO) oxidation was measured simultaneously in a group of five normal subjects after an oral load of 100 g naturally-labelled [13C]glucose, using indirect calorimetry and mass spectrometry. 2. CHO utilization, calculated from the results of indirect calorimetry, increased 30 min after the glucose load to reach a peak at 90 min. It then decreased to reach basal values at 380 min. Cumulative total CHO oxidation at 480 min was 83±8g, and CHO oxidized above basal levels, 37±3 g. 3. Enrichment of expired carbon dioxide with 13C began at 60 min and maximum values were observed at 270 min. At 480 min, cumulative CHO oxidation measured by use of [13C]glucose was 29 g. The difference from calorimetric values can be attributed in part to the slow isotopic dilution in the glucose and bicarbonate pools. 4. Thus, approximately 30% of the glucose load was oxidized during the 8 h after its ingestion and this accounts for a significant part of the increased CHO oxidation (37 g), as measured by indirect calorimetr

Reflection and Ducting of Gravity Waves Inside the Sun

Internal gravity waves excited by overshoot at the bottom of the convection zone can be influenced by rotation and by the strong toroidal magnetic field that is likely to be present in the solar tachocline. Using a simple Cartesian model, we show how waves with a vertical component of propagation can be reflected when traveling through a layer containing a horizontal magnetic field with a strength that varies with depth. This interaction can prevent a portion of the downward-traveling wave energy flux from reaching the deep solar interior. If a highly reflecting magnetized layer is located some distance below the convection zone base, a duct or wave guide can be set up, wherein vertical propagation is restricted by successive reflections at the upper and lower boundaries. The presence of both upward- and downward-traveling disturbances inside the duct leads to the existence of a set of horizontally propagating modes that have significantly enhanced amplitudes. We point out that the helical structure of these waves makes them capable of generating an alpha-effect, and briefly consider the possibility that propagation in a shear of sufficient strength could lead to instability, the result of wave growth due to over-reflection.Comment: 23 pages, 5 figures. Accepted for publication in Solar Physic