Flux Law Research Articles

Ice‐Tethered Profiler observations of the double‐diffusive staircase in the Canada Basin thermocline

Six Ice‐Tethered Profilers (ITP), deployed in the central Canada Basin of the Arctic Ocean between 2004 and 2007, have provided detailed potential temperature and salinity measurements of a double‐diffusive staircase at about 200–300 m depth. Individual layers in the staircase are of order 1 m in vertical height but appear to extend horizontally for hundreds of kilometers, with along‐layer gradients of temperature and salinity tightly related. On the basis of laboratory‐derived double‐diffusive flux laws, estimated vertical heat fluxes through the staircase are in the range 0.05–0.3 W m−2, only about one tenth of the estimated mean surface mixed layer heat flux to the sea ice. It is thus concluded that the vertical transport of heat from the Atlantic Water in the central basin is unlikely to have a significant impact to the Canada Basin ocean surface heat budget. Icebreaker conductivity‐temperature‐depth data from the Beaufort Gyre Freshwater Experiment show that the staircase is absent at the basin periphery. Turbulent mixing that presumably disrupts the staircase might drive greater flux from the Atlantic Water at the basin boundaries and possibly dominate the regionally averaged heat flux.

A Consistent Description of Scaling Law for Flux Pinning in ${\hbox{Nb}}_{3}{\hbox{Sn}}$ Strands Based on the Kramer Model (Corrected)*

There are a lot of experimental reports on the scaling of flux pinning in the form of F = F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> (1 - b) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , with b = B/B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</sub> .The temperature dependence of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> is approximately proportional to B'.2 , whereas the strain dependence of Fm is reported to be proportional to the upper critical field Bc2. In this work, we re-analyze our previous data with the Kramer model including the pin-breaking dynamic pinning force (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ) for a low field region. It is shown that the extrapolated upper critical field B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</sub> *, strongly depend on the ratio between the mean of the parameter K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> for F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> (<K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> >) and the parameter K, for the flux line lattice shearing pinning force F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub> . It is found that the strain dependence of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> at 4.2 K is approximately proportional to (B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</sub> *) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> . We further compare the data with the prediction of our recent scaling theory based on Eliashberg theory of strongly coupled superconductors. It is shown that the strain dependence of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> at 4.2 K is proportional to BC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5/2</sup> kappa <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> , consistent with the temperature dependence of Fm. Moreover, this model agrees reasonably well even with the data in a high compressive strain region (<-0.8%).

Nonlocal heat transfer in nanostructures

Kinetics of electrons in a degenerate conductor heated up by absorption of a high-frequency field localized in a region of about hundred nanometers has been studied. A new law for nonlocal electron thermal flux has been predicted.

Fourier's Law for a Granular Fluid

Newton's viscosity law for the momentum flux and Fourier's law for the heat flux define Navier-Stokes hydrodynamics for a simple, one-component fluid. There is ample evidence that a hydrodynamic description applies as well to a mesoscopic granular fluid with the same form for Newton's viscosity law. However, theory predicts a qualitative difference for Fourier's law with an additional contribution from density gradients even at uniform temperature. The reasons for the absence of such terms for normal fluids are indicated, and a related microscopic explanation for their existence in granular fluids is presented.

Convergence to equilibrium for a parabolic–hyperbolic phase field model with Cattaneo heat flux law

In this paper we consider the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase field system: (0.1) { χ t − Δ χ + χ 3 − χ − θ = 0 , θ t + χ t + div q = 0 , q t + q + ∇ θ = 0 , in Ω × ( 0 , + ∞ ) subject to the homogeneous Neumann boundary condition for χ, (0.2) ∂ n χ = 0 , on Γ × ( 0 , + ∞ ) , and no-heat flux boundary condition for q, (0.3) q ⋅ n = 0 , on Γ × ( 0 , + ∞ ) , and the initial conditions (0.4) χ ( 0 ) = χ 0 , θ ( 0 ) = θ 0 , q ( 0 ) = q 0 , in Ω , where Ω ⊂ R 3 is a bounded domain with a smooth boundary Γ and n is the outward normal direction to the boundary. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity.

A mixed type identification problem related to a phase-field model with memory

In this paper we consider an integro-differential system consisting of a parabolic and a hyperbolic equation related to phase transition models. The first equation is integro-differential and of hyperbolic type. It describes the evolution of the temperature and also accounts for memory effects through a memory kernel $k$ via the Gurtin-Pipkin heat flux law. The latter equation, governing the evolution of the order parameter, is semilinear, parabolic and of the fourth order (in space). We prove a local in time existence result and a global uniqueness result for the identification problem consisting in recovering the memory kernel $k$ appearing in the first equation.

Convection under a lid of finite conductivity: Heat flux scaling and application to continents

A scaling law for the heat flux out of a convective fluid covered totally or partially by a finitely conducting lid is proposed. This scaling is constructed in order to quantify the heat transfer out of the Earth's mantle, taking into account the effect of the dichotomy between oceans and continents, which imposes heterogeneous thermal boundary conditions at the surface of the mantle. The effect of these heterogeneous boundary conditions is studied here using simple two‐dimensional models, with the mantle represented by an isoviscous fluid heated from below and continents represented by nondeformable lids of finite thermal conductivity set above the surface of the model. We use free‐slip boundary conditions under the oceanic and continental zones in order to study in an isolated way the possible thermal effect of continents, independently of all mechanical effect. A systematic study of the heat transfer as a function of the Rayleigh number of the fluid, of the width of the lid, and of its thermal properties is carried out. We show that estimates of continental lithosphere thickness imply a strong insulating effect from continents on mantle heat loss, at least locally. The heat flux below continents was low in the past and of the order of the present one if the continental thickness has remained broadly constant over the Earth's history.

Growth, decay and bifurcation of shock amplitudes under the type-II flux law

By replacing Fick's diffusion law with Green and Nagdhi's type-II flux law, a hyperbolic counterpart to the classical Fisher–KPP equation is obtained. In this article, an analytical study of this partial differential equation is presented with an emphasis on shock and related kinematic wave phenomena. First, an exact travelling wave solution (TWS) is derived and examined. Then, using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained. In addition, the issue of shock stability is addressed and the limitations of the model are noted. It is shown that discontinuity (i.e. shock) formation in the TWS occurs only when the propagation speed, which must exceed the characteristic speed, tends to the latter. It is also shown that the shock amplitude equation undergoes a transcritical bifurcation. Lastly, numerical simulations of acceleration waves in a simple model problem are presented.

Mechanics of merging events for a series of layers in a stratified turbulent fluid

This study attempts to explain the evolutionary pattern of a series of well-mixed layers separated by thin high-gradient interfaces frequently observed in stratified fluids. Such layered structures form as a result of the instability of the equilibrium with uniform stratification, and their subsequent evolution is characterized by a sequence of merging events which systematically increase the average layer thickness. The coarsening of layers can take one of two forms, depending on the realized vertical buoyancy flux law. Layers merge either when the high-gradient interfaces drift and collide, or when some interfaces gradually erode without moving vertically. The selection of a preferred pattern of coarsening is rationalized by the analytical theory – the merging theorem – which is based on linear stability analysis for a series of identical layers and strongly stratified interfaces. The merging theorem suggests that the merger by erosion of weak interfaces occurs when the vertical buoyancy flux decreases with the buoyancy variation across the step. If the buoyancy flux increases with step height, then coarsening of a staircase may result from drift and collision of the adjacent interfaces. Our model also quantifies the time scale of merging events and makes it possible to predict whether the layer merging continues indefinitely or whether the coarsening is ultimately arrested. The merging theorem is applied to extant one-dimensional models of turbulent mixing and successfully tested against the corresponding fully nonlinear numerical simulations. It is hypothesized that the upscale cascade of buoyancy variance associated with merging events may be one of the significant sources of the fine-scale (∼ 10m) variability in the ocean.

Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations

We study the geometry and dynamics of both isolated and dynamical trapping horizons by considering the allowed variations of their foliating two-surfaces. This provides a common framework that may be used to consider both their possible evolutions and their deformations as well as derive the well-known flux laws. Using this framework, we unify much of what is already known about these objects as well as derive some new results. In particular we characterize and study the ``almost isolated'' trapping horizons known as slowly evolving horizons. It is for these horizons that a dynamical first law holds and this is analogous and closely related to the Hawking-Hartle formula for event horizons.

Responses of soil‐mantled hillslopes to transient channel incision rates

Channel incision drives hillslope morphology in humid soil‐mantled landscapes. When channel incision rates change, numerous hillslope soil properties (e.g., erosion rates, soil thickness, and soil production) adjust in response to this change. Here we investigate the timescales of adjustment of hillslope soil properties when channel incision rates change in time. An idealized soil‐mantled hillslope (linear sediment flux law and soil density equal to bedrock density) is investigated, and the transient evolution of this hillslope is determined analytically. This analysis reveals two dominant adjustment timescales. The longer of these two timescales determines the rate at which the entire hillslope adjusts to a change in channel incision rates and is proportional to 4λ2/(π2D), where λ is the length of the hillslope and D is the sediment diffusivity. Numerical simulations are then used to examine responses of hillslopes that involve nonlinear behavior (e.g., hillslopes that experience nonlinear sediment transport or have soil thicknesses that respond to the soil production function). Using these numerical models, we show that the ratio between the soil density (ρs) and the density of the soil parent material (ρη) can alter the long‐term response of the hillslope such that the characteristic timescale becomes 4ρηλ2/(π2Dρs). In addition, we show that the adjustments of the soil erosion rate, the soil production rate, and the soil thickness have different characteristic response timescales. Hillslopes that experience sediment flux that is proportional to the depth slope product respond on longer timescales than hillslopes that experience a linear sediment flux law when channel incision rates increase. We illustrate how the spatial pattern of hillslope response to changes in channel incision rates can be used to constrain either channel incision histories or hillslope response timescales. If the hillslope response timescale is known, the pattern of hillslope disturbance can be used to constrain the celerity of a incision wave as it moves upstream through a channel. If the channel incision history is known, the hillslope response timescale may be evaluated on the basis of the spatial pattern of hillslope disturbance.

Non-isothermal, non-Newtonian Hele–Shaw flows within Cattaneo’s heat flux law

In this paper, we consider a non-isothermal, non-Newtonian injection process. This leads to the study of a novel elliptic–hyperbolic system. The hyperbolic nature of the system arises because we replace the infinite speed of propagation from classical, thermal elasticity by a finite propagation velocity. We present a formal derivation of the elliptic–hyperbolic system starting from conservations of mass, momentum, and energy in a three-dimensional domain, where the removal of the infinite propagation speed is achieved using Cattaneo’s law for heat conduction. The existence of weak solutions to certain elliptic–hyperbolic problems associated with the resulting equations is proved.

Asymptotic behaviour for a two‐dimensional thermoelastic model

AbstractIn this paper we study a thermoelastic material with an internal structure which binds the materials fibres to a quadratic behaviour. Moreover, a hereditary constitutive law for heat flux is supposed. We prove results of asymptotic stability and exponential decay for the evolution problem in two‐dimensional space domain. Copyright © 2006 John Wiley &amp; Sons, Ltd.

The Growth of Ice Crystals by Molecular Diffusion

Abstract The mass transfer of water molecules by diffusion onto ice particles is best described by their Sherwood number (Sh), a dimensionless quantity, which combines molecular and convective effects and depends on the airflow as represented by the Reynolds number (Re). While Sh (Re &amp;gt; 0) has been previously measured in experiments for typical crystal shapes, the limiting case of pure molecular diffusion (Sh0) for zero flow with Re = 0 is not known well and needs independent determination. The direct numerical solution of the controlling Laplace equation links diffusion with electric fields through the electrostatic analogy. It will be solved for the electrostatic potential V around a crystal-shaped conductor of capacitance C. The results will then be converted by similarity theory. This led to the first numerical determination of Sh0 for hexagonal plates, hexagonal columns, stellar crystals, capped columns, and broad-branched crystals. The new data represent another necessary step in the formulation of an experiment-based theory of the growth of freely falling ice crystals in the atmosphere. A discrete version of Gauss's flux law is developed to compute the flux generated by a crystal-shaped conductor in a finite Cartesian grid box, using a Gauss–Seidel iterative scheme. This method is general and can be applied to compute Sh0 for any rectilinear shapes to any degree of accuracy. The dimensionless mass transfer by molecular diffusion, Sh0, is identical to the diffusion of heat characterized by the Nusselt number Nu0.

Mathematical Models of Reissner–Mindlin Thermoviscoelastic Plates

In this paper we investigate mathematical models describing deformations and thermal variations of a thin homogeneous thermoviscoelastic plate. A hereditary non-Fourier constitutive law for the heat flux and some heat power constitutive equation with linear memory are considered. The resulting models are derived in the framework of the well-established theory, due to Gurtin and Pipkin, and according to the standard approximation procedure for the Reissner–Mindlin plate model.

GZK cutoff distortion due to the energy error distribution shape

The development of an ultra high energy air shower has an intrinsic energy fluctuation due both to the first interaction point and to the cascade development. Here we show that for a given primary energy this fluctuation has a lognormal distribution and thus observations will estimate the primary energy with a lognormal error distribution. We analyze the UHECR energy spectrum convolved with the lognormal energy error and demonstrate that the shape of the error distribution will interfere significantly with the ability to observe features in the spectrum. If the standard deviation of the lognormal error distribution is equal or larger than 0.25, both the shape and the normalization of the measured energy spectra will be modified significantly. As a consequence, the GZK cutoff might be sufficiently smeared as not to be seen (without very high statistics). This result is independent of the power law of the cosmological flux. As a conclusion we show that in order to establish the presence or not of the GZK feature, not only more data are needed but also that the shape of the energy error distribution has to be known well. The high energy tail and the sigma of the approximate lognormal distribution of the error in estimating the energy must be at the minimum set by the physics of showers.

Review of Water Flow in Soils, 2nd Ed., by Tsuyoshi MiyazakiWater Flow in SoilsTaylor &amp; Francis Group

This book is composed of nine chapters. The first six chapters and the last chapter are written by Miyazaki; Chapter 7 is co-authored with K. Seki, and Chapter 8 with S. Hasegawa and T. Kasubuchi. For an authored book this seems an unusual arrangement. Introducing the system of units employed in the book, Chapter 1 discusses the structure of soils and the retentivity of water therein. The structure includes both solid phase and soil pores, as well as liquid phase and gas phase in soil pores. Matric potential, soil moisture characteristic curves, osmotic effect on retentitivity, and hysteresis encompass the material on the retentivity of water in soils. Chapter 2, containing five parts, discusses physical laws of water flow in soils. Introducing Darcy’s law for water flux in one, two, and three dimensions, the first part goes on to present basic equations governing the flow of water in soils, unsaturated hydraulic conductivity and empirical relations thereof, and soil water diffusivity without any mathematical relations. The second part discusses the gas-phase configuration in unsaturated flow, including free and entrapped air in soils, forced-closed system, water flow in open systems, and significance of gas-phase configuration in fields. The discussion is comprehensive and has a lot of significance in irrigated agriculture and hydrologic modeling, as well as solute transport. The third part is on infiltration and discusses basic aspects of infiltration and infiltration into dry soils and wet soils. Mathematical equations of infiltration included are Horton, Holtan, Green–Ampt, and Philip. There are other useful empirical and conceptual equations, such as Kostyakov, SCS-CN, Singh-Yu, Smith, Parlange, and others, that have been found to be useful in irrigation engineering and watershed modeling, but have not been included. Steady flows, both upward and downward, are discussed in Part IV. The last part deals with transient flow in soils, including evaporation, redistribution, and water flow in deep soils. The chapter is quite good but would have been stronger if it had a discussion on stochasticity and reliability of different formulations. A discussion of errors and limitations of different formulations is helpful to the reader. Refraction of water flow in soils is the subject matter of Chapter 3. It encompasses refraction of flux, theory of refraction, jump condition, visualization of refraction in layered soils, and verification of refraction theory experimentally as well as numerically. Refraction and anisotropy coefficient in saturated layered and unsaturated layered soils, and refractive anisotropy coefficient in two-layered and multilayered models, are also treated in the chapter. This provides a comprehensive treatment of refraction theory. Chapter 4 deals with preferential flow. Beginning with a classification of flow, it goes on to discuss general features; vertical and lateral bypassing flow; criteria and hydrodynamics of fingering flow; and nature, experimental observation, qualitative features, and modeling of funneled flow. Both saturated and

Lateral migration of hillcrests in response to channel incision in soil‐mantled landscapes

We investigate lateral migration of hillcrests in response to vertical offsets or transient incision rates of channels bordering these hillcrests in soil‐mantled landscapes. For hillslopes undergoing sediment transport that is linearly proportional to the slope, the hillcrest offset distance is one quarter of the ratio of the vertical offset between the channels to the relief of the symmetric hillslope. If channels are downcutting at different rates, the speed of hillcrest migration will depend on the ratio of the downcutting rates and the density ratio, which is the ratio of the bulk density of the bedrock to the bulk density of the soil. The density ratio plays a fundamental role in determining the transient response of the hillcrest; lower‐density ratios lead to faster transient responses to changes in channel downcutting rates. Other parameters that affect the transient response of the hillcrest are the magnitude of transient differences in downcutting between the two channels, the time‐averaged incision rate, and a ratio of the elevation of the hillslope to a length that characterizes the decay in soil production with increasing soil thickness; different parameters will be important for different sediment flux laws. The profile of soil thickness reacts to transient changes in downcutting at a different rate than surface topography. Hillslopes experiencing transient channel downcutting may have surface topography that is symmetric about the hillcrest but will at the same time have a soil thickness profile that is asymmetric.

Well-Posedness and Regularity for a Parabolic-Hyperbolic Penrose-Fife Phase Field System

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable χ, which is characterized by the presence of an inertial term multiplied by a small positive coefficient µ. This feature is the main consequence of supposing that the response of χ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature ϑ, i.e. α(ϑ) ∼ ϑ − 1/ϑ. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as µ ↘ 0. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.

Growth and decay of shock and acceleration waves in a traffic flow model with relaxation

The usual Ficken-based constitutive relation for traffic flux is replaced with one based on the Maxwell–Cattaneo model. The resulting flux law, which now takes into account the reaction time of driver and vehicle, results in a second-order, hyperbolic generalization of Burgers’ equation as the PDE governing the traffic density. An analytical study of this equation is presented with an emphasis on shock and related kinematic wave phenomena. Specifically, the exact traveling wave solution (TWS) is derived and it is shown that shock formation is possible only if the diffusivity is non-vanishing. Using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained and their temporal evolution determined. The exact upper bound of the reaction time parameter is established and connections between singular surface and TWS results are noted.