One-dimensional Advection-diffusion Model Research Articles

Modeling of brain efflux: Constraints of brain surfaces

Fluid efflux from the brain plays an important role in solute waste clearance. Current experimental approaches provide little spatial information, and data collection is limited due to short duration or low frequency of sampling. One approach shows tracer efflux to be independent of molecular size, indicating bulk flow, yet also decelerating like simple membrane diffusion. In an apparent contradiction to this report, other studies point to tracer efflux acceleration. We here develop a one-dimensional advection-diffusion model to gain insight into brain efflux principles. The model is characterized by nine physiological constants and three efflux parameters for which we quantify prior uncertainty. Using Bayes' rule and the two efflux studies, we validate the model and calculate data-informed parameter distributions. The apparent contradictions in the efflux studies are resolved by brain surface boundaries being bottlenecks for efflux. To critically test the model, a custom MRI efflux assay measuring solute dispersion in tissue and release to cerebrospinal fluid was employed. The model passed the test with tissue bulk flow velocities in the range 60 to 190 [Formula: see text]m/h. Dimensional analysis identified three principal determinants of efflux, highlighting brain surfaces as a restricting factor for metabolite solute clearance.

Simulation of bidisperse colloidal centrifugal sedimentation using a mixture viscosity model

Understanding the sedimentation behavior of bidisperse colloidal suspensions is critical in determining their stability and separation. While centrifugation is often used to accelerate separation, the settling of bidisperse colloids and their phase separation under these conditions is complex and difficult to predict explicitly. As an alternative, this work proposes a one-dimensional advection-diffusion model that uses an effective maximum volume fraction with a bidisperse viscosity scheme, which reflects important characteristics of bidisperse sedimentation while remaining computationally efficient. The influence of Derjaguin–Landau–Verwey–Overbeek interactions on packing fraction and dispersion viscosity is also considered. A numerical implementation is described using an adaptive finite-difference solver, which can be used for concentration profile and settling rate prediction of both species under variable acceleration. Validation experiments with silica suspensions in two size ratios (500:800 and 100:500 nm) and various total concentrations are performed using an analytical centrifuge, with results also being compared to Richardson–Zaki empirical predictions. The model is shown to be a very good fit to the data for both size ratio dispersions at three mixing ratios, with differences <10%. Slightly higher levels of variation were detected for the 500:800 nm system, owing to the smaller size ratio and resulting greater effect of uncounted secondary hydrodynamic factors, which enables the limits of the mixture viscosity model to be established. Nevertheless, this work highlights that mixture viscosity modeling combined with effective maximum volume fraction modifications can provide critical insights into the effect of bidisperse suspension dynamics on separation efficiencies.

Hyperbolic advection–diffusion schemes for high-Reynolds-number boundary-layer problems

This paper discusses issues in hyperbolic advection–diffusion schemes for high-Reynolds-number boundary-layer problems. Implicit hyperbolic advection–diffusion solvers have been found to encounter significant convergence deterioration for high-Reynolds-number problems with boundary layers. The problems are examined in details for a one-dimensional advection–diffusion model, and resolutions are discussed. One of the major findings is that the relaxation length scale needs to be inversely proportional to the Reynolds number to minimize the truncation error and retain the dissipation of the hyperbolic diffusion scheme. Accurate, robust, and efficient boundary-layer calculations by hyperbolic schemes are demonstrated for advection–diffusion equations in one and two dimensions.

Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium

This work presents the analytical solution and temporal moments of one-dimensional advection–diffusion model with variable coefficients. Two case studies along with the two different sets of boundary conditions are considered at the inlet and outlet of the domain. In the first case, a time-dependent solute dispersion in the homogeneous domain along uniform flow is taken into account, whereas in the second case, due to inhomogeneity of domain, velocity is taken spatially dependent and the dispersion is assumed proportional to the square of the velocity. The Laplace transform is used to obtain the analytical solutions. The analytical temporal moments are derived from the Laplace domain solutions. To verify the correctness of the analytical solutions, a high-resolution second-order finite volume scheme is applied. Different case studies are considered and discussed. Both analytical and numerical results are in good agreement with each other.

The Path Density of Interhemispheric Surface-to-Surface Transport. Part I: Development of the Diagnostic and Illustration with an Analytic Model

Abstract A new path-density diagnostic for atmospheric surface-to-surface transport is formulated. The path density η gives the joint probability that air whose last surface contact occurred on patch Ωi at time ti will make its next surface contact with patch Ωf after a residence time τ ∈ (τ, τ + dτ) and that it can be found in d3r during its surface-to-surface journey. The dependence on τ allows the average surface-to-surface flow rate carried by the paths to be computed. A simple algorithm for using passive tracers to determine η is developed. A key advantage of the diagnostic is that it can be computed efficiently without an adjoint model and using only a moderately large number of tracers. The nature of the path density is illustrated with a one-dimensional advection–diffusion model. In Part II of this study, the path density diagnostic is applied to quantify interhemispheric transport through the troposphere and stratosphere.

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection–diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α , with 1 < α ⩽ 2 . We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection–diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.

O2 consumption rate and its isotopic fractionation factor in the deep water of the Philippine Sea

Concentration and stable isotopic compositions (δ18O) of dissolved O2 were measured in seawater samples collected from the Philippine Sea in June 2006. The in-situ O2 consumption rate and the isotopic fractionation factor (αr) during dissolved O2 consumption were obtained from field observations by applying a vertical one-dimensional advection diffusion model to the deep water mass of about 1000–4000 m. The average O2 consumption rate and αr were, respectively, 0.11 ± 0.07 µmol kg−1yr−1 and 0.990 ± 0.001. These estimated values agree well with values from earlier estimations of Pacific deep water. The in-situ O2 consumption rates are two or more times higher north of 20°N, although the value of αr was not significantly different between the north and south. Its levels varied rapidly in the water mass of less about 2000 m depth. These results suggest that organic matter from the continent imparts a meaningful contribution to the upper water in the northern part of the area; it might produce the strong O2 minimum that is evident in the water mass from about 1000–2000 m in the northern part of the Philippine Sea.

Criteria for resolution-scales and parameterisation of compartmental models of hydrological and ecological mass flows

The effect of compartment resolution-scale, i.e. number of compartments and compartment size, on prediction of transport of water and solute mass in watersheds is analysed by using numerical discretisation and temporal moments methods. The one-dimensional advection–diffusion model is used as a theoretical frame of reference for representing transport along a single series of compartments along a pathway in the surface water or ecosystem. Criteria for parameterisation of compartment models and error estimates are derived for two cases: (1) requiring an exact match with the advection–diffusion model; and (2) requiring as good match as possible with a simplified compartment model using a small number of compartments. The matching criteria are based on a comparison of numerical discretisation and temporal moments of the compartmental backflow model with those of the one-dimensional advection–diffusion equation. We derived solutions in terms of temporal moments of both the advection–diffusion equation and compartmental backflow model with arbitrary input concentration vs. time boundary condition. This approach can be used to translate parameters between the models and analyse the error caused by any arbitrary structure of the compartmental model as a function of boundary conditions and the physical parameters of the advective–diffusive equation. Data from tracer experiments were compared with model predictions to provide a practical frame of reference.

Distribution of natural and man-made radionuclides during the reoccupation of GEOSECS stations 413 and 416 in the Arabian Sea: temporal changes

Measurement of natural and man-made 14 C and 3 H and man-made 90 Sr, 239,240 Pu and 241 Am at two GEOSECS stations, GX 413 and 416 reoccupied two decades later are reported. Deeper penetration of nutrient and oxygen-poor intermediate waters along with man-made radionuclides, especially at GX 413 (near Bab el Mandab), are noticeable. The one-dimensional advection-diffusion model using pre-nuclear 14 C data supports this observation as the vertical advection velocity at GX 413 is 11.8 m/y compared to the 3.6 m/y in the central north Arabian Sea (GX 416).

Cold bottom water events observed in the Hawaii Ocean Time-series: implications for vertical mixing

During 1991–1999 six near-bottom, cold water anomaly events occurred at the Hawaii Ocean Time-series deep-ocean hydrographic station (22°45′N, 158°W), situated on the northern flank of a seafloor basin known as the Kauai Deep. Cold anomalies of at least 0.01°C, and as much as 0.024°C, were observed in near-bottom CTD profiles during each event. The anomalies relaxed towards more typical conditions over a period of several months. On the basis of water mass properties, we infer that the cold anomalies in the Kauai Deep originated when colder, saltier water from the adjacent Maui Deep spilled over the sill that separates these abyssal basins. The cause of these cold overflow events is not yet known, but inverted echo sounder observations and TOPEX/Poseidon altimeter data suggest that they might be associated with eddies entering the region of the Maui Deep. The transport across the sill and the vertical structure of the eddy diffusivity parameter were estimated with a one-dimensional advection–diffusion model, considering the evolving shape of the anomalous potential temperature profile during the third event onset and the apparent overflow duration. The best subjective fit to the observations yields a transport across the sill of 0.18 Sv, and a constant vertical eddy diffusivity of 7×10 −4 m 2 s −1. The observations during the recovery of the anomalous temperatures for four of the events were used to calculate the terms of a one-dimensional diffusion model, yielding the vertical and temporal variability of the turbulent diffusivity. The results show a consistent picture of the eddy diffusivity reaching median values of 40×10 −4 m 2 s −1 near the region of the sill depth and decreasing above and below it. This maximum increase seems to be due to enhanced levels of turbulence near the depth of the controlling sill. The diffusivities also peak at the beginning of the relaxation period, decreasing by one order of magnitude by the end of each event. The fifth event's overflow was apparently captured in progress with two full-depth CTD profiles obtained 46 h apart, during which the near-bottom temperature decreased 0.035°C, the salinity increased 0.003, and the dissolved oxygen increased 7 μmol kg −1. A seiching motion or deep baroclinic basin mode resulting from the event could explain this and other large differences in near-bottom temperatures observed between deep casts taken a few days apart.

Constant ventilation age of thermocline water in the eastern subtropical North Pacific Ocean from chlorofluorocarbon measurements over a 12‐year period

Northeastern Pacific chlorofluorocarbon (CFC) data collected between 1982 and 1994 near Geochemical Ocean Sections Study (GEOSECS) station 1 (28.5°N, 122.5°W) record decadal timescale ventilation processes of the subtropical thermocline in this region. The CFC‐12 concentration age field versus potential density has been remarkably constant over the 12‐year period, although CFC concentrations in the upper kilometer of the water column have increased with time. Results from a simple one‐dimensional advection‐diffusion model are consistent with an advection velocity of ca. 0.8–0.9 cm s−1 from the source area. The influence of the 1982–83 El Niño is noticeable in the 1983 observations. While the main stream of subarctic source water appears to spread southward at a constant rate, during El Niño years the influence of comparatively CFC‐free tropical thermocline waters is enhanced in this region, leading to reduced vertical inventories of CFCs, but without changing the apparent CFC ages as functions of potential density. Apparent oxygen utilization rates decrease with increasing CFC age, and also appear not to have changed significantly over the 12‐year measurement period.

Dynamics of landfill gas migration in unconsolidated sands

A gas plume from the Foxhall landfill in Suffolk, UK contains elevated concentrations of methane and carbon dioxide in unsaturated unconsolidated sands which comprise Pleistocene “Red Crag” deposits. The plume emanates from a relatively restricted zone along one side of the landfill, but extends over 100 m from the site boundary. The reduction in the methane to carbon dioxide ratio with distance and the systematic changes in their carbon and hydrogen stable isotope ratios are evidence of microbially mediated methane oxidation. A one-dimensional advection–diffusion model was used to describe the combined concentration of methane and carbon dioxide in the plume. Diffusion alone underestimated the concentration profile, but a good fit to the data was achieved with an advective flux of 4.5 m yr–1, indicating that advection due to a pressure gradient from the landfill as well as diffusion should be considered in gas migration modelling. The kinetics of methane oxidation was studied by parameter-fitting a reaction rate into the advection–diffusion equation with first-order decay. A decay constant of –0.063 yr–1 (half-life 11 yr) produced a poor fit to the methane profile, suggesting that oxidation may not be constant throughout the plume. However, the stable isotope data allowed two rates of oxidation to be inferred. A slow rate of oxidation with a half-life of the order of 4.3 to 7.6 yr was inferred in the centre of the plume where oxygen was absent. A much faster rate with a half-life no longer than 0.76 to 1.21 yr occurred beyond 60 m of the landfill and around the top fringe of the plume where oxygen was present. These rates are considered to reflect the difference between aerobic and anaerobic oxidation, the latter utilizing iron(III) in the sediment as an electron acceptor. The shape of the plume is asymmetrical, indicating a geological control on gas migration. In a two-dimensional model a poor fit to the observed data was obtained when the sand was assumed to be homogeneous and where the gas entered from a restricted part of the landfill boundary. However, a better model was produced by varying the diffusion coefficient in the sands over the range 5 × 10–7 to 2 × 10–6 m2 s–1 without the need to restrict the zone of gas release along the landfill boundary. Such a range in transport properties could be accounted for by normal variability in the porosity, tortuosity and water content of the sand. The long-term dissipation of the plume assuming only diffusion was predicted to take up to 30 yr for the gas concentration to reduce to 10% of its initial value. However, the plume disappeared within a year after pumping from gas wells in the landfill, indicating that advection under an imposed pressure gradient was a major control on remediation. This study shows that models can be used to explain landfill gas migration and to infer oxidation rates which can be used to predict gas migration at other sites. However, the need to obtain field data on gas permeabilities and diffusivities will always be a major limitation in predicting gas migration in permeable formations.

Long-term variations of potential temperature and dissolved oxygen of the Japan Sea Proper Water

Observed potential temperatures and concentrations of dissolved oxygen are analyzed to elucidate their variations during the period from 1958 to 1996 at Stn. P (37°43′ N, 134°43′ E) and from 1965 to 1996 at Stn. H (40°30′ N, 137°40′ E) in the Japan Sea. At Stn. P, increases of the potential temperature for the period are found below 800 m depth with the largest value of 0.16 ± 0.09°C per century at 800 m depth. At Stn. H, the potential temperature increased below 500 m depth. The increase rate has the largest value of 0.50 ± 0.18°C per century at 500 m depth and it is 0.30 ± 0.09°C per century at 800 m depth. The concentrations of dissolved oxygen increased around 800 m depth at Stn. P. At Stn. H, they increased above 800 m depth. On the other hand, they decreased below 1200 m depth at both stations. The layer of the dissolved oxygen minimum has deepened in these decades. These features appearing in the distributions of temperature and dissolved oxygen are successively simulated by a vertical one-dimensional advection-diffusion model including consumption of dissolved oxygen and termination of the deep water supply. These results suggest that the supply of the Japan Sea Proper Water into the deep layer, which is cold and rich in dissolved oxygen, has been decreasing for the last four decades.

Estimation of vertical mixing in the upper ocean at Station P from chlorofluorcarbons

Vertical mixing (K v ) in the upper ocean is estimated from chlorofluorocarbons (CFC's) F-11 and F-12 data collected on 5 cruises (1982, 1985, 1992, 1993, 1995) near Station P (50N, 145W). A one-dimensional advection-diffusion model was fitted to the data from each cruise to estimate vertical mixing (K v ) and upwelling velocity (w). With constant K v and w, the average value of K v and w was 0.4 ± 0,1 cm 2 s -1 and 1.2 ± 0.4 m yr -1 respectively for the depth range 0-900 m below the base of the mixed layer. This case produced K v values that increased with time, and modeled CFC concentrations that were higher than observed in the upper increased and lower than observed in the deeper water (200-900 m). Both of these conditions are consistent with K v values that increased with depth. Fitting the one-dimensional advection-diffusion model to the data with K v inversely related to the buoyancy frequency reduced the model-data misfit by 40%, produced consistent estimates of K v the all cruises and reduced the systematic diffrences in the model data misfits. From this model K v and w at the base of the mixed layer were 0.15 ±.04 cm 2 s -1 and 0.5 ± 0.15 m yr -1 , respectively. The results strongly supported a K v that increased with depth. Modeled anthropogenic CO 2 penetration at Station P with the inverse buoyancy frequency scaling of K v , produced results consistent with the observed anthropogenic CO 2 penetration inferred from Σ CO 2 , alkalinity and apparent oxygen utilization measurements.

The influence of spatially and temporally varying oceanographic conditions on meroplanktonic metapopulations

Abstract We synthesize the results of several modelling studies that address the influence of variability in larval transport and survival on the dynamics of marine metapopulations distributed along a coast. Two important benthic invertebrates in the California Current System (CCS), the Dungeness crab and the red sea urchin, are used as examples of the way in which physical oceanographic conditions can influence stability, synchrony and persistence of meroplanktonic metapopulations. We first explore population dynamics of subpopulations and metapopulations. Even without environmental forcing, isolated local subpopulations with density-dependence can vary on time scales roughly twice the generation time at high adult survival, shifting to annual time scales at low survivals. The high frequency behavior is not seen in models of the Dungeness crab, because of their high adult survival rates. Metapopulations with density-dependent recruitment and deterministic larval dispersal fluctuate in an asynchronous fashion. Along the coast, abundance varies on spatial scales which increase with dispersal distance. Coastwide, synchronous, random environmental variability tends to synchronize these metapopulations. Climate change could cause a long-term increase or decrease in mean larval survival, which in this model leads to greater synchrony or extinction respectively. Spatially managed metapopulations of red sea urchins go extinct when distances between harvest refugia become greater than the scale of larval dispersal. All assessments of population dynamics indicate that metapopulation behavior in general dependes critically on the temporal and spatial nature of larval dispersal, which is largely determined by physical oceanographic conditions. We therfore explore physical influences on larval dispersal patterns. Observed trends in temperature and salinity applied to laboratory-determined responses indicate that natural variability in temperature and salinity can lead to variability in larval development period on interannual (50%), intra-annual (20%) and latitudinal (200%) scales. Variability in development period significantly influences larval survival and, thus, net transport. Larval drifters that undertake diel vertical migration in a primitive equation model of coastal circulation (SPEM) demonstrate the importance of vertical migration in determining horizontal transport. Empirically derived estimates of the effects of wind forcing on larval transport of vertically migrating larvae (wind drift when near the surface and Ekman transport below the surface) match cross-shelf distributions in 4 years of existing larval data. We use a one-dimensional advection-diffusion model, which includes intra-annual timing of cross-shelf flows in the CCS, to explore the combined effects on settlement: (1) temperature- and salinity-dependent development and survival rates and (2) possible horizontal transport due to vertical migration of crab larvae. Natural variability in temperature, wind forcing, and the timing of the spring transition can cause the observed variability in recruitment. We conclude that understanding the dynamics of coastally distributed metapopulations in response to physically-induced variability in larval dispersal will be a critical step in assessing the effects of climate change on marine populations.

228Ra and its implications in the Seto Inland Sea

The features of 228Ra and 226Ra distribution covering the salinity range from less than 0·1% to 35% in the Ota River estuary and extending to the Seto Inland Sea have been presented. Using this data and a mass balance of radium in this region, the source of radium from the river-estuarine sediment is shown to be more important than that from the sediments themselves of the Seto Inland Sea. The flux of Ra from the sediments is 0·84 dpm 228Ra cm -2 year -1 and 0·08 dpm 226Ra cm -2 year -1 about 17 times higher than those from river for 228Ra and 8 times for 226 Ra. By applying the One-Dimensional Advection-Diffusion model and assuming a steady state of radium in the Seto Inland Sea, the appropriate value of the horizontal eddy diffusion coefficient is in the range 1·1 × 10 6 - 1·4 × 106 cm 2 s -1.

Chlorofluoromethanes in South Atlantic Antartic intermediate water

Distribution of the dissolved atmospheric chlorofluoromethanes (CFMs) F-11 and F-12 in the South Atlantic Ocean are used to study the ventilation and circulation of Antarctic Intermediate Water (AAIW). CFM distributions on an isopycnal surface representative of AAIW are consistent with recently ventilated water entering the subtropical gyre in the southwestern Atlantic and then being advected anticyclonically around this gyre. The westward-flowing northern limb of the gyre apparently divides near the coast of South America with some water flowing southward to recirculate in the gyre, and the balance flowing northward along the coast of Brazil. At the equator this northward current divides again with one branch going eastward along the equator and the other continuing into the Northern Hemisphere. In the eastern tropical Atlantic, the CFM concentrations on this isopycnal surface in the cyclonic gyre are extremely low between the subtropical gyre and the equatorial tongue. Along the prime meridian, the F-11 and F-12 concentrations on the 27.2 θσ isopycnal surface between the mixed layer outcrop and the northern edge of the subtropical gyre are fitted to a one-dimensional advection-diffusion model. This model assumes that the CFMs enter the tubtropical gyre solely by northward diffusion from the mixed layer outcrop to the southern edge of the subtropical gyre, and that their distributions within the gyre are controlled by both advective and diffusive processes. Velocity and eddy diffusion coefficients are calculated from a least-squares fit to the data. These values are then used to calculate a mean oxygen consumption rate which is consistent with rates calculated using models of other time-dependent geochemical tracers.

Dynamics of dissolved organic carbon in the northwestern Indian Ocean

We report here the non-conservative behaviour of DOC in the northwestern Indian Ocean by studying this parameter together with other carbon and nitrogen components. This contrasts with earlier reports of conservative behaviour. Concentrations of DOC, 3–4 times higher than those reported earlier, were found to decrease northward from the equator. Total carbon dioxide (TCO 2) increases in proportion of the oxygen utilized, thus revealing the dominant biological role in the carbon turnover. The CO 2 added through dissolution of biogenic debris is found to decrease southward, in general. Decomposition of organic material contributes at least 64% to the CO 2 addition that increases southward, the rest being from dissolution of skeletal material. Evidence is provided for the utilization of oxygen and nitrate for DOC oxidative decomposition. Accumulation of DOC without its complete oxidation to CO 2 could be the main reason for the TCO 2 decrease in southern Arabian Sea. Relationships of DOC with nitrification and denitrification processes show that the microbial population plays a major role in regulating the DOC contents in the seawater of this region. Consumption/decomposition by denitrifying bacteria and other micro-organisms responsible for nitrogen cycling in the sea are found to be intimately related to the DOC dynamics and are responsible for decreased DOC concentrations in the north. DOC accumulation in the southern Arabian Sea seems to facilitate bacterio-particulate aggregate formation and consequent nitrification, which results in excess nitrate. Application of a one-dimensional advection-diffusion model to the present data set provides evidence for the non-conservative nature of DOC in the Arabian Sea.

Dynamics of dissolved organic carbon in the northwestern Indian Ocean

We report here the non-conservative behaviour of DOC in the northwestern Indian Ocean by studying this parameter together with other carbon and nitrogen components. This contrasts with earlier reports of conservative behaviour. Concentrations of DOC, 3–4 times higher than those reported earlier, were found to decrease northward from the equator. Total carbon dioxide (TCO 2 ) increases in proportion of the oxygen utilized, thus revealing the dominant biological role in the carbon turnover. The CO 2 added through dissolution of biogenic debris is found to decrease southward, in general. Decomposition of organic material contributes at least 64% to the CO 2 addition that increases southward, the rest being from dissolution of skeletal material. Evidence is provided for the utilization of oxygen and nitrate for DOC oxidative decomposition. Accumulation of DOC without its complete oxidation to CO 2 could be the main reason for the TCO 2 decrease in southern Arabian Sea. Relationships of DOC with nitrification and denitrification processes show that the microbial population plays a major role in regulating the DOC contents in the seawater of this region. Consumption/decomposition by denitrifying bacteria and other micro-organisms responsible for nitrogen cycling in the sea are found to be intimately related to the DOC dynamics and are responsible for decreased DOC concentrations in the north. DOC accumulation in the southern Arabian Sea seems to facilitate bacterio-particulate aggregate formation and consequent nitrification, which results in excess nitrate. Application of a one-dimensional advection-diffusion model to the present data set provides evidence for the non-conservative nature of DOC in the Arabian Sea.