Steady-state Width Research Articles

Spatiotemporal velocities, aggregation, and false dispersion: numerical considerations for the modeling of colonial and motile harmful algae

PurposeThe purpose of this study is to investigate the errors arising from the numerical treatment of model processes, paying particular attention to the impact of key system features including widely variable dispersion coefficients, spatiotemporal velocities of algal cells, and the aggregation of algae from single cells to large colonies. An advection–dispersion model has been presented to describe the vertical transport of colonial and motile harmful algae in a lake environment.Design/methodology/approachModel performance is examined for two different numerical treatments of the advective term: first-order upwind and quadratic upwind with a stability-preserving flux limiter (SMART). To determine how these schemes impact predictions, comparisons are made across a sequence of models with increasing complexity.FindingsUsing first-order upwinding for advection–dispersion calculations with a time oscillating velocity field leads to oscillatory numerical dispersion. Subjecting an initially uniform distribution of large-sized algal colonies to a spatiotemporal velocity creates a concentration pulse, which reaches a steady-state width at high-grid Peclet numbers when using the SMART scheme; the pulse exhibits contraction–expansion behavior throughout a velocity cycle at all Peclet numbers when using first-order upwinding. When aggregation dynamics are included with advection-dominated spatiotemporal transport, results indicate the SMART scheme predicts larger peak concentration values than those predicted by first-order upwind, but peak location and the time to large colony appearance remain largely unchanged between the two advective schemes.Originality/valueTo the best of the authors’ knowledge, this study is the first numerical investigation of a novel advection–dispersion model of vertical algal transport. In addition, a generalized expression for the effective dispersion coefficient of temporally variable flow fields is presented.

Assessing the role of thermal disequilibrium in the evolution of the lithosphere–asthenosphere boundary: an idealized model of heat exchange during channelized melt transport

Abstract. This study explores how the continental lithospheric mantle (CLM) may be heated during channelized melt transport when there is thermal disequilibrium between (melt-rich) channels and surrounding (melt-poor) regions. Specifically, I explore the role of disequilibrium heat exchange in weakening and destabilizing the lithosphere from beneath as melts infiltrate into the lithosphere–asthenosphere boundary (LAB) in intraplate continental settings. During equilibration, hotter-than-ambient melts would be expected to heat the surrounding CLM, but we lack an understanding of the expected spatiotemporal scales and how these depend on channel geometries, infiltration duration, and transport rates. This study assesses the role of heat exchange between migrating material in melt-rich channels and their surroundings in the limit where advective effects are larger than diffusive heat transfer (Péclet numbers > 10). I utilize a 1D advection–diffusion model that includes thermal exchange between melt-rich channels and the surrounding melt-poor region, parameterized by the volume fraction of channels (ϕ), average relative velocity (vchannel) between material inside and outside of channels, channel spacing (d), and timescale of episodic or repeated melt infiltration (τ). The results suggest the following: (1) during episodic infiltration of hotter-than-ambient melt, a steady-state thermal reworking zone (TRZ) associated with spatiotemporally varying disequilibrium heat exchange forms at the LAB. (2) The TRZ grows by the transient migration of a disequilibrium-heating front at a material-dependent velocity, reaching a maximum steady-state width δ proportional to ϕvchannel(τ/d)n, where n≈2 for periodic thermal perturbations and n≈1 for a single finite-duration thermal pulse. For geologically reasonable model parameters, the spatiotemporal scales associated with establishment of the TRZ are comparable with those inferred for the migration of the LAB based on geologic observations within continental intra-plate settings, such as the western US. The results of this study suggest that, for channelized transport speeds of vchannel=1 m yr−1, channel spacings d≈102 m, and timescales of episodic melt infiltration τ≈101 kyr, the steady-state width of the TRZ in the lowermost CLM is ≈10 km. (3) Within the TRZ, disequilibrium heat exchange may contribute ≈10-5 W m−3 to the LAB heat budget.

Stream Spreading in Multilayer Microfluidic Flows of Suspensions

The goal of this experimental study is to quantify the spreading of parallel streams with viscosity contrast in multilayer microfluidic flows. Three streams converge into one channel where a test fluid is sheathed between two layers of a Newtonian reference fluid. The test fluids are Newtonian fluids with viscosities ranging from 1.1 to 48.2 cP and suspensions of 10-mum-diameter PMMA particles with particle volume fractions phi = 0.16-0.30. The fluid interface locations are identified through fluorescence microscopy. The steady-state width of the center stream is strongly dependent on the viscosity ratio between the adjacent fluids and exhibits a near power-law relationship. This dependence occurs for both the Newtonian fluids and the suspensions, although the slopes differ. The high-concentration suspension (phi = 0.30) diverges from Newtonian behavior, while the low-concentration suspensions (phi = 0.16, 0.22) closely approximate that of the Newtonian fluids. The observed suspension behavior can be attributed to shear-induced particle migration.

A Coupled Coastal Polynya–Atmospheric Boundary Layer Model

Abstract This paper formulates and presents opening solutions of a one-dimensional coastal polynya flux model in which frazil ice is characterized by its depth and concentration. In comparison with polynya flux models in which variable frazil ice concentration is absent, this model is found to predict a smaller heat flux to the atmosphere. Consequently the model in this paper exhibits a wider steady-state polynya and longer opening times when compared with models in which ice concentration is neglected. The aforementioned polynya flux model is then coupled to a lower-atmosphere boundary layer model, and it is demonstrated that the polynya opening time and the steady-state width are significantly altered in the coupled, as compared with the decoupled, system. In essence, the heating of the lower atmosphere above the evolving polynya in the coupled system reduces the sensible heat flux between the ocean and atmosphere, thereby reducing the frazil ice production rate and hence leading to longer polynya opening time and wider steady-state width. This phenomenon is particularly noticeable when the potential temperature of the atmosphere at the coast is only slightly below the freezing point. In addition, a cutoff atmospheric wind speed is shown to exist, above which a steady-state polynya can never be obtained. Solutions calculated by the two models, using parameters representative of the St. Lawrence Island polynya, show that the new models contain substantial predictive capability.

Interface relaxation in electrophoretic deposition of polymer chains: effects of segmental dynamics, molecular weight, and field.

Using different segmental dynamics and relaxation, characteristics of the interface growth is examined in an electrophoretic deposition of polymer chains on a three (2+1)-dimensional discrete lattice with a Monte Carlo simulation. Incorporation of faster modes such as crankshaft and reptation movements along with the relatively slow kink-jump dynamics seems crucial in relaxing the interface width. As the continuously released polymer chains are driven (via segmental movements) and deposited, the interface width W grows with the number of time steps t, W proportional, variant t(beta), (beta approximately 0.4-0.8), which is followed by its saturation to a steady-state value W(s). Stopping the release of additional chains after saturation while continuing the segmental movements relaxes the saturated width to an equilibrium value (W(s)-->W(r)). Scaling of the relaxed interface width W(r) with the driving field E, W(r) proportional, variant E(-1/2) remains similar to that of the steady-state W(s) width. In contrast to monotonic increase of the steady-state width W(s), the relaxed interface width W(r) is found to decay (possibly as a stretched exponential) with the molecular weight.

Width distributions and the upper critical dimension of Kardar-Parisi-Zhang interfaces.

Simulations of restricted solid-on-solid growth models are used to build the width distributions of d=2-5 dimensional Kardar-Parisi-Zhang (KPZ) interfaces. We find that the universal scaling function associated with the steady-state width distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity.

High-intensity nanosecond photorefractive spatial solitons

We report the observation of high-intensity solitons in a bulk strontium barium niobate crystal. The solitons are observed by use of 8-ns optical pulses with optical intensities greater than 100 MW/cm(2). Each soliton forms and attains its minimum width after roughly ten pulses and reaches e(-1) of the steady-state width after the first pulse. We find good agreement between experimental observations and theoretical predictions for the soliton existence curve.

Singular effects of surface tension in evolving Hele-Shaw flows

In this paper, we present evidence to show that a smoothly evolving zero-surface tension solution of the Hele-Shaw equations can be singularly perturbed by the presence of arbitrarily small non-zero surface tension in order-one time. These effects are explained by the impact of ‘daughter singularities’ on the physical interface, whose formation was suggested in a prior paper (Tanveer 1993). For the case of finger motion in a channel, it is seen that the daughter singularity effect is strong enough to produce the transition from a finger of arbitrary width to one with the selected steady-state width in O(1) time.

Steady states for viscous fingers with anisotropic surface tension

Pattern selection is considered for the case of viscous fingering in rectangular Hele-Shaw geometry in the presence of anisotropic surface tension, using solvability analysis and a boundary-integral method. We find that anisotropy introduced as a sinusoidal perturbation with a fourfold symmetry is irrelevant for small driving velocities and the usual steady-state finger width in the absence of the anisotropy is obtained. For sufficiently large driving velocities a new steady-state width is selected when the anisotropy makes the local interfacial tension a maximum at the finger tip. This is in agreement with recent experimental observations.

Width distribution for (2+1)-dimensional growth and deposition processes.

Nonequilibrium growth processes are frequently characterized by the width w(L,t) of the active zone, where t is the time elapsed since the start of the process and L is the spatial interval over which the measurement is carried out. Quite generally, w(L,t) obeys a scaling form w(L,t)\ensuremath{\sim}${\mathit{L}}^{\mathrm{\ensuremath{\zeta}}}$f(${\mathit{tL}}^{\mathrm{\ensuremath{-}}\mathit{z}}$), and many workers have attempted to determine the dynamic universality class of such processes by a measurement of the exponents \ensuremath{\zeta} and z. In this paper, we calculate the steady-state width distribution P(${\mathit{w}}^{2}$) for several three-dimensional growth processes and show that, expressed in a suitable form, P(${\mathit{w}}^{2}$) can be used to distinguish between different possible universality classes. We also reanalyze experimental data obtained by scanning-tunneling or atomic-force microscopy and show that P(${\mathit{w}}^{2}$) provides valuable information on the nature of a growth process.

Factors controlling the width of shear lips in fatigue crack growth

An analysis is made of the factors which control the width of the shear lips which are found on fatigue crack surfaces of aluminium alloy sheet material. Tests on 100-mm wide plates of 2024-T351 Al alloy in air showed that if cracks were grown at constant ΔK the shear lips reached a steady-state width which depended only on ΔK ϵff , After the beginning of a test it takes a few mm of crack growth before an equilibrium state is reached. The maximum rate of widening of shear lips depends on K μax . Most constant-amplitude (ΔS) tests do not show the steady-state shear lip width. Based on the results of the tests at constant ΔK, a method of calculating the width of shear lips as a function of a or ΔK in a constant-amplitude test is presented. The calculated values have been compared with the results from constant-amplitude tests.

Subglacial Water flow Under Ice Streams and West Antarctic Ice-Sheet Stability

An analysis is made of the steady-state width of an ice sheet whose base is below sea-level, whose basal temperatures are such that appreciable melting occurs at the base, and which is fringed by fastmoving ice streams that drain most of the outward ice flux. The fast ice velocities of the ice streams are considered to be a consequence of substantial subglacial water flow underneath the ice streams. The source of this water is the water melted from the base of the ice sheet which is diverted to flow beneath the ice streams. If the depth of the sea at the edge of the ice sheet is not a function of the width of the ice sheet, then an ice sheet with a steady-state width is in a situation of unstable equilibrium. Only if the sea-level depth at the edge of the ice sheet increases as a function of ice-sheet width at a rate greater than the 2/3rd power of the width can a stable, steady-state ice sheet exist. This condition (taking into account elastic rebound) is not satisfied for the West Antarctic ice sheet along an ice-flow path from the ice divide above Byrd station out to the Ross Sea. An increase of the mean precipitation, such as might occur under a C02-induced climatic warming, would cause growth of both stable or unstable steady-state ice sheets.

Subglacial Water flow Under Ice Streams and West Antarctic Ice-Sheet Stability

An analysis is made of the steady-state width of an ice sheet whose base is below sea-level, whose basal temperatures are such that appreciable melting occurs at the base, and which is fringed by fastmoving ice streams that drain most of the outward ice flux. The fast ice velocities of the ice streams are considered to be a consequence of substantial subglacial water flow underneath the ice streams. The source of this water is the water melted from the base of the ice sheet which is diverted to flow beneath the ice streams. If the depth of the sea at the edge of the ice sheet is not a function of the width of the ice sheet, then an ice sheet with a steady-state width is in a situation of unstable equilibrium. Only if the sea-level depth at the edge of the ice sheet increases as a function of ice-sheet width at a rate greater than the 2/3rd power of the width can a stable, steady-state ice sheet exist. This condition (taking into account elastic rebound) is not satisfied for the West Antarctic ice sheet along an ice-flow path from the ice divide above Byrd station out to the Ross Sea. An increase of the mean precipitation, such as might occur under a C02-induced climatic warming, would cause growth of both stable or unstable steady-state ice sheets.

Plasma Leakage Through a Low-βLine Cusp

The half-width of the leakage aperture of plasma through a low-$\ensuremath{\beta}$ line cusp is found to be twice the hybrid gyroradius ${({r}_{e}{r}_{i})}^{\frac{1}{2}}$ with ${r}_{e}$ and ${r}_{i}$ as the electron and ion gyroradii. This steady-state width is shown to develop in a time less than the ion gyroperiod.