Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

The advection-condensation model and water-vapour probability density functions

The statistically steady humidity distribution resulting from an interaction of advection, modelled as an uncorrelated random walk of moist parcels on an isentropic surface, and a vapour sink, modelled as immediate condensation whenever the specific humidity exceeds a specified saturation humidity, is explored with theory and simulation. A source supplies moisture at the deep-tropical southern boundary of the domain and the saturation humidity is specified as a monotonically decreasing function of distance from the boundary. The boundary source balances the interior condensation sink, so that a stationary spatially inhomogeneous humidity distribution emerges. An exact solution of the Fokker-Planck equation delivers a simple expression for the resulting probability density function (PDF) of the wate-rvapour field and also the relative humidity. This solution agrees completely with a numerical simulation of the process, and the humidity PDF exhibits several features of interest, such as bimodality close to the source and unimodality further from the source. The PDFs of specific and relative humidity are broad and non-Gaussian. The domain-averaged relative humidity PDF is bimodal with distinct moist and dry peaks, a feature which we show agrees with middleworld isentropic PDFs derived from the ERA interim dataset. Copyright (C) 2011 Royal Meteorological Society

Design and Validation of On-chip Planar Mixer Based on Advection and Viscoelastic Effects

Mixing at low Reynolds number is usually due to diffusion and requires longer channel lengths for complete mixing. In order to reduce the mixing lengths, advective flow can be induced by varying the channel geometry. Additionally, in non-newtonian fluids, appropriate modifications to channel geometry can be used to aid the mixing process by capitalizing on their viscoelastic nature. Here we have exploited the advection and viscoelastic effects to implement a planar passive micro-mixer. Microfluidic devices incorporating different blend of mixing geometries were conceived. The optimum design was chosen based on the results of the numerical simulations performed in COMSOL. The chosen design had sudden expansion and contraction along with teeth patterns along the channel walls to improve mixing. Mixing of two different dyes was performed to validate the mixing efficiency. Particle dispersion experiments were also carried out. The results indicated effective mixing. In addition, the same design was also found to be compatible with electrical power free pumping mechanism like suction. The proposed design was then used to carry out on-chip chemical cell lysis with human whole blood samples to establish its use with non-newtonian fluids. Complete lysis of the erythrocytes was observed leaving behind the white blood cells at the outlet.

Resuspension and advection processes affecting suspended particulate matter concentrations in the central English Channel

Suspended particulate matter (SPM) measurements obtained along a cross-section in the central English Channel (Wight-Cotentin transect) indicate that the area may be differentiated into: (1) an English coastal zone, associated with the highest concentrations; (2) a French coastal zone, with intermediate concentrations; and (3) the offshore waters of the Channel, characterised by a very low suspended-sediment load. The SPM particle-size distribution was modal close to the English coast (main mode 10-12 mu m); the remainder of the area was characterised by flat SPM distributions. Examination of the diatom communities in the SPM suggest:; that material resuspended in the intertidal zone and the estuarine environments was advected towards the offshore waters of the English Channel. Considerable variations in SPM concentrations occurred during a tidal cycle: maximum concentrations were sometimes up to 3 times higher than the minimum concentrations, Empirical orthogonal function (EOF) analysis of the SPM concentration time series indicates that, although the bottom waters were more turbid than the surficial waters, this was not likely to be the result of in situ sediment resuspension. Instead, the observed variations appear to be controlled mainly by advective mechanisms. The limited resuspension was probably caused by: (1) the limited availability of fine-grained material within the bottom sediments, and (2) 'bed-armouring' processes which protect the finer-grained fractions of the seabed material from erosion and entrainment within the overlying flow during the less energetic stages of the tide.

Interannual horizontal heat advection in the surface mixed layer over the equatorial Pacific Ocean: assimilation versus TAO analysis

The interannual anomalies of horizontal heat advection in the surface mixed layer over the equatorial Pacific Ocean in an assimilation experiment are studied and compared with existing observational analyses. The assimilation builds upon a hindcast study that has produced a good simulation of the observed equatorial currents and optimizes the simulation of the Reynolds sea surface temperature (SST) data. The comparison suggests that the assimilation has improved the simulation of the interannual horizontal heat advection of the surface mixed layer significantly. During periods of interrupted current measurements, the assimilation is shown to produce more meaningful anomalies of the heat advection than the interpolation of the observational data does. The assimilation also shows that the eddy heat flux due to the correlation between high-frequency current and SST variations, which is largely overlooked by the existing observational analyses, is important for the interannual SST balance over the equatorial Pacific. The interannual horizontal heat advection anomalies are found to be sensitive to SST errors where oceanic currents are strong, which is a challenge for ENSO prediction. The study further suggests that the observational analyses of the tropical SST balance based on the TAO and the Reynolds SST data contain significant errors due to the large gradient errors in the Reynolds SST data, which are amplified into the advection anomalies by the large equatorial currents.

Boundary layer growth and advection of heat over snow and soil patches: Modelling and parametrization

Essery, RLH, RJ Granger and JW Pomeroy, 2006. Boundary layer growth and advection of heat over snow and soil patches: Modelling and parametrization. Hydrological Processes, 20, 953 - 967.

The role of advection in the distribution of plankton populations at a moored 1-D coastal observatory

The degree to which advection modulates the distribution of plankton populations at a 1-D coastal observatory was assessed at station L4 in the western English Channel (50°15′N 4°13′W, depth 50 m), part of the Western Channel Observatory (WCO). Five tidal-cycle surveys were conducted, three in spring and two in summer 2010. Observations of the physical characteristics of L4 were obtained by using a moored acoustic doppler current profiler (ADCP) and a free-falling microstructure sensor (MSS). The moored ADCP highlighted the presence of vertical shear, with typical values of U during spring tides of ∼0.5 m s−1 at the surface and ∼0.2 m s−1 at the bed. The distribution of phyto- and zooplankton populations above a size threshold of 200 μm were examined using an in-line holographic imaging system, the Holocam. Variability in time as well as depth is a common feature throughout each of the surveys, with examples of recorded numbers of phytoplankton that ranged between 1300 L−1 and 2300 L−1 at the same depth but at different points within the tidal cycle. Further, at the same points in the tidal cycle the number of recorded zooplankton was also seen to vary, specifically with the identification of gelatinous planula in spring that increased the observed number to maximums of between 140 L−1 and 220 L−1 in the upper layer, considerably higher that the corresponding WP-2 net counts for a similar period. Specific aspects of the movement and transfer of plankton relating to advection and interaction with the pycnocline are identified, both across tidal cycles and seasons.

Solving the advection-dispersion equation using the discrete puff particle method

A flexible, mass-conservative numerical technique for solving the advection-dispersion equation for miscible contaminant transport is presented. The method combines features of puff transport models from air pollution studies with features from the random walk particle method used in water resources studies, providing a deterministic time-marching algorithm which is independent of the grid Peclet number and scales from one to higher dimensions simply. The concentration field is discretised into a number of particles, each of which is treated as a point release which advects and disperses over the time interval. The dispersed puff is itself discretised into a spatial distribution of particles whose masses can be pre-calculated. Concentration within the simulation domain is then calculated from the mass distribution as an average over some small volume. Comparison with analytical solutions for a one-dimensional fixed-duration concentration pulse and for two-dimensional transport in an axisymmetric flow field indicate that the algorithm performs well. For a given level of accuracy the new method has lower computation times than the random walk particle method.

Illustration of micro-scale advection using grid-pattern mini-lysimeters

Twenty-five small soil-filled perspex boxes arranged in a square, with dwarf sunflowers growing in them, were used to study micro-scale advection. Hydrological heterogeneity was introduced by applying two different amounts of irrigation water (low-irrigation, L, versus high-irrigation, H). The nine central boxes (4 H, 4 L and I bare box) were precision weighing lysimeters, yielding diurnal measurements of evaporation. After the onset of soil water stress, a large difference in latent heat flux (up to 4-fold) was observed between the lysimeters of the H and L treatments, mainly caused by large differences between H and L stomatal conductance values. This resulted in micro-advection, causing H soil-sunflower systems to evaporate well above equilibrium latent heat flux. The occurrence of micro-advective enhancement was reflected in large values of the Priestley-Taylor constant (often larger than 2.0) and generally negative values of sensible heat flux for the H treatment. (c) 2005 Elsevier B.V. All rights reserved.