arXiv subjects

John S. Wettlaufer

Publications and source records attributed to John S. Wettlaufer.

At least 37 records · Page 2Linked to original sources

arXiv2017

Stochastic dynamics of Arctic sea ice Part II: Multiplicative noise

We analyze the numerical solutions of a stochastic Arctic sea ice model with multiplicative noise over a wide range of external heat-fluxes, $ΔF_0$, which correspond to greenhouse gas forcing. When the noise is multiplicative, the noise-magnitude depends on the state-variable, and this will influence the statistical moments in a manner that differs from the additive case, which we analyzed in Part I of this study. The state variable describing the deterministic backbone of our model is the energy, $E(t)$, contained in the ice or the ocean and for a thorough comparison and contrast we choose the simplest form of multiplicative noise $σE(t) ξ(t)$, where $σ$ is the noise amplitude and $ξ(t)$ is the noise process. The case of constant additive noise (CA) we write as $σ\overline{E_S}ξ(t)$, in which $\overline{E_S}$ is the seasonally averaged value of the periodic deterministic steady-state solution $E_S(t)$, or the deterministic seasonal cycle. We then treat the case of seasonally-varying additive noise (SVA), $σE_S(t) ξ(t)$, as well as two types of multiplicative noise that depend on the form of stochastic calculus (Itô or Stratonovich) used to interpret the noise itself. The comparison of these four cases reveals the stochastic anatomy of the system over the entire range of the $ΔF_0$ from the perennial ice states to near the ice-free state.

physics.ao-ph↗

arXiv2017

Stochastic dynamics of Arctic sea ice Part I: Additive noise

We analyze the numerical solutions of a stochastic Arctic sea ice model with constant additive noise over a wide range of external heat-fluxes, $ΔF_0$, which correspond to greenhouse gas forcing. The variability that the stochasticity provides to the deterministic steady state solutions (perennial and seasonal ice states) is illustrated by examining both the stochastic paths and probability density functions (PDFs). The principal stochastic moments (standard deviation, mean and skewness) are calculated and compared with those determined from a stochastic perturbation theory described previously by Moon and Wettlaufer (2013). We examine in detail the competing roles of the destabilizing sea ice-albedo-feedback and the stabilizing long-wave radiative loss contributions to the variability of the ice cover under increased greenhouse-gas forcing. In particular, the variability of the stochastic paths at the end of summer shows a clear maximum, which is due to the combination of the increasing importance of the albedo-feedback and its associated "memory effect" that accumulates from early spring to late summer. This is counterbalanced by the stabilization of the ice cover due to the longwave loss of energy from the ice surface, which is enhanced during winter, thereby focusing the stochastic paths and decreasing the variability in winter. The organizing principle for the interpretation of these effects is a generalized Ornstein-Uhlenbeck process with a time dependent potential.

physics.ao-ph↗

arXiv2017

The statistical properties of sea ice velocity fields

By arguing that the surface pressure field over the Arctic Ocean can be treated as an isotropic, stationary, homogeneous, Gaussian random field, Thorndike estimated a number of covariance functions from two years of data (1979 and 1980). Given the active interest in changes of general circulation quantities and indices in the polar regions during the recent few decades, the spatial correlations in sea ice velocity fields are of particular interest. It is thus natural to ask; "how persistent are these correlations?" To this end, a multi-fractal stochastic treatment is developed to analyze observed Arctic sea ice velocity fields from satellites and buoys for the period 1978 - 2015. Having previously found that the Arctic Equivalent Ice Extent (EIE) has a white noise structure on annual to bi-annual time scales, the connection between EIE and ice motion is assessed. The long-term stationarity of the spatial correlation structure of the velocity fields, and the robustness of their white noise structure on multiple time scales is demonstrated, which (a) combine to explain the white noise characteristics of the EIE on annual to biannual time scales, and (b) explain why the fluctuations in the ice velocity are proportional to fluctuations in the geostrophic winds on time scales of days to months. Moreover, it is shown that the statistical structure of these two quantities is commensurate from days up to years, which may be related to the increasing prevalence of free drift in the ice pack.

physics.ao-ph↗

arXiv2017

Roughness as a Route to the Ultimate Regime of Thermal Convection

We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for $Pr = 1$ and $Ra = \left[4 \times 10^6, 3 \times 10^9\right]$. By varying the wavelength $λ$ at a fixed amplitude, we find an optimal wavelength $λ_{\text{opt}}$ for which the Nusselt-Rayleigh scaling relation is $\left(Nu-1 \propto Ra^{0.483}\right)$ maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering \cite{Goluskin:2016} who prove that $Nu$ can grow no faster than ${\cal O} (Ra^{1/2})$ as $Ra \rightarrow \infty$, and thus the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When $λ\ll λ_{\text{opt}}$ and $λ\gg λ_{\text{opt}}$, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each $Ra$ we choose the maximum $Nu$ among all $λ$, and thus optimizing over all $λ$, to find $Nu_{\text{opt}} - 1 = 0.01 \times Ra^{0.444}$.

physics.flu-dyn↗

arXiv2016

Statistical Mechanics and the Climatology of the Arctic Sea Ice Thickness Distribution

We study the seasonal changes in the thickness distribution of Arctic sea ice, $g(h)$, under climate forcing. Our analytical and numerical approach is based on a Fokker-Planck equation for $g(h)$ (Toppaladoddi \& Wettlaufer \emph{Phys. Rev. Lett.} {\bf 115}, 148501, 2015), in which the thermodynamic growth rates are determined using observed climatology. In particular, the Fokker-Planck equation is coupled to the observationally consistent thermodynamic model of Eisenman \& Wettlaufer (\emph{Proc. Natl. Acad. Sci. USA} {\bf 106}, pp. 28-32, 2009). We find that due to the combined effects of thermodynamics and mechanics, $g(h)$ spreads during winter and contracts during summer. This behavior is in agreement with recent satellite observations from CryoSat-2 (Kwok \& Cunningham, \emph{Phil. Trans. R. Soc. A} {\bf 373}, 20140157, 2015). Because $g(h)$ is a probability density function, we quantify all of the key moments (e.g., mean thickness, fraction of thin/thick ice, mean albedo, relaxation time scales) as greenhouse-gas radiative forcing, $ΔF_0$, increases. The mean ice thickness decays exponentially with $ΔF_0$, but {\em much slower} than do solely thermodynamic models. This exhibits the crucial role that ice mechanics plays in maintaining the ice cover, by redistributing thin ice to thick ice--far more rapidly than can thermal growth alone.

physics.ao-ph↗

arXiv2016

Surface tension and a self-consistent theory of soft composite solids with elastic inclusions

The importance of surface tension effects is being recognized in the context of soft composite solids, where it is found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length $L\equivγ/E$, where $γ$ is the inclusion/host surface tension and $E$ is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length $L$, the spherical inclusion radius $R$, and the stiffness contrast parameter ${C}$, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.

cond-mat.soft↗

arXiv2016

Exoplanetary Detection By Multifractal Spectral Analysis

Owing to technological advances, the number of exoplanets discovered has risen dramatically in the last few years. However, when trying to observe Earth analogs, it is often difficult to test the veracity of detection. We have developed a new approach to the analysis of exoplanetary spectral observations based on temporal multifractality, which identifies time scales that characterize planetary orbital motion around the host star, and those that arise from stellar features such as spots. Without fitting stellar models to spectral data, we show how the planetary signal can be robustly detected from noisy data using noise amplitude as a source of information. For observation of transiting planets, combining this method with simple geometry allows us to relate the time scales obtained to primary and secondary eclipse of the exoplanets. Making use of data obtained with ground-based and space-based observations we have tested our approach on HD 189733b. Moreover, we have investigated the use of this technique in measuring planetary orbital motion via Doppler shift detection. Finally, we have analyzed synthetic spectra obtained using the SOAP 2.0 tool, which simulates a stellar spectrum and the influence of the presence of a planet or a spot on that spectrum over one orbital period. We have demonstrated that, so long as the signal-to-noise-ratio $\ge$ 75, our approach reconstructs the planetary orbital period, as well as the rotation period of a spot on the stellar surface.

astro-ph.EP↗

arXiv2016

Large deformations of a soft porous material

Compressing a porous material will decrease the volume of the pore space, driving fluid out. Similarly, injecting fluid into a porous material can expand the pore space, distorting the solid skeleton. This poromechanical coupling has applications ranging from tissue mechanics to hydrogeology. The classical theory of linear poroelasticity captures this coupling by combining Darcy's law with Terzaghi's effective stress and linear elasticity in a linearized kinematic framework. Linear poroelasticity is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. The well-known theory of large-deformation poroelasticity combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework. This theory has been used extensively in biomechanics to model large elastic deformations in soft tissues, and in geomechanics to model large elastoplastic deformations in soils. Here, we first provide an overview and discussion of this theory with an emphasis on the physics of poromechanical coupling. We present the large-deformation theory in an Eulerian framework to minimize the mathematical complexity, and we show how this nonlinear theory simplifies to linear poroelasticity under the assumption of small strain. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of two uniaxial model problems: Fluid outflow driven by an applied mechanical load (the consolidation problem) and compression driven by a steady fluid throughflow. We use these problems to explore the steady and dynamical errors associated with the linear model in both situations, as well as the impact of introducing a deformation-dependent permeability.

physics.flu-dyn↗

arXiv2015

Surface tension and the Mori-Tanaka theory of non-dilute soft composite solids

Eshelby's theory is the foundation of composite mechanics, allowing calculation of the effective elastic moduli of composites from a knowledge of their microstructure. However it ignores interfacial stress and only applies to very dilute composites -- i.e. where any inclusions are widely spaced apart. Here, within the framework of the Mori-Tanaka multiphase approximation scheme, we extend Eshelby's theory to treat a composite with interfacial stress in the non-dilute limit. In particular we calculate the elastic moduli of composites comprised of a compliant, elastic solid hosting a non-dilute distribution of identical liquid droplets. The composite stiffness depends strongly on the ratio of the droplet size, $R$, to an elastocapillary length scale, $L$. Interfacial tension substantially impacts the effective elastic moduli of the composite when $R/L\lesssim 100$. When $R < 3L/2$ ($R=3L/2$) liquid inclusions stiffen (cloak the far-field signature of) the solid.

cond-mat.soft↗

arXiv2015

Interfacial tension and a three-phase generalized self-consistent theory of non-dilute soft composite solids

In the dilute limit Eshelby's inclusion theory captures the behavior of a wide range of systems and properties. However, because Eshelby's approach neglects interfacial stress, it breaks down in soft materials as the inclusion size approaches the elastocapillarity length $L$. Here, we use a three-phase generalized self-consistent method to calculate the elastic moduli of composites comprised of an isotropic, linear-elastic compliant solid hosting a spatially random monodisperse distribution of spherical liquid droplets. As opposed to similar approaches, we explicitly capture the liquid-solid interfacial stress when it is treated as an isotropic, strain-independent surface tension. Within this framework, the composite stiffness depends solely on the ratio of the elastocapillarity length $L$ to the inclusion radius $R$. Independent of inclusion volume fraction, we find that the composite is stiffened by the inclusions whenever $R < 3L/2$. Over the same range of parameters, we compare our results with alternative approaches (dilute and Mori-Tanaka theories that include surface tension). Our framework can be easily extended to calculate the composite properties of more general soft materials where surface tension plays a role.

cond-mat.soft↗

arXiv2015

Tailoring boundary geometry to optimize heat transport in turbulent convection

By tailoring the geometry of the upper boundary in turbulent Rayleigh-Bénard convection we manipulate the boundary layer -- interior flow interaction, and examine the heat transport using the Lattice Boltzmann method. For fixed amplitude and varying boundary wavelength $λ$, we find that the exponent $β$ in the Nusselt-Rayleigh scaling relation, $Nu-1 \propto Ra^β$, is maximized at $λ\equiv λ_{\text{max}} \approx (2 π)^{-1}$, but decays to the planar value in both the large ($λ\gg λ_{\text{max}}$) and small ($λ\ll λ_{\text{max}}$) wavelength limits. The changes in the exponent originate in the nature of the coupling between the boundary layer and the interior flow. We present a simple scaling argument embodying this coupling, which describes the maximal convective heat flux.

physics.flu-dyn↗

arXiv2015

Fluid-driven deformation of a soft granular material

Compressing a porous, fluid-filled material will drive the interstitial fluid out of the pore space, as when squeezing water out of a kitchen sponge. Inversely, injecting fluid into a porous material can deform the solid structure, as when fracturing a shale for natural gas recovery. These poromechanical interactions play an important role in geological and biological systems across a wide range of scales, from the propagation of magma through the Earth's mantle to the transport of fluid through living cells and tissues. The theory of poroelasticity has been largely successful in modeling poromechanical behavior in relatively simple systems, but this continuum theory is fundamentally limited by our understanding of the pore-scale interactions between the fluid and the solid, and these problems are notoriously difficult to study in a laboratory setting. Here, we present a high-resolution measurement of injection-driven poromechanical deformation in a system with granular microsctructure: We inject fluid into a dense, confined monolayer of soft particles and use particle tracking to reveal the dynamics of the multi-scale deformation field. We find that a continuum model based on poroelasticity theory captures certain macroscopic features of the deformation, but the particle-scale deformation field exhibits dramatic departures from smooth, continuum behavior. We observe particle-scale rearrangement and hysteresis, as well as petal-like mesoscale structures that are connected to material failure through spiral shear banding.

cond-mat.soft↗

arXiv2015

On the nature of the sea ice albedo feedback in simple models

We examine the nature of the ice-albedo feedback in a long standing approach used in the dynamic-thermodynamic modeling of sea ice. The central issue examined is how the evolution of the ice area is treated when modeling a partial ice cover using a two-category-thickness scheme; thin sea ice and open water in one category and "thick" sea ice in the second. The problem with the scheme is that the area-evolution is handled in a manner that violates the basic rules of calculus, which leads to a neglected area-evolution term that is equivalent to neglecting a leading-order latent heat flux. We demonstrate the consequences by constructing energy balance models with a fractional ice cover and studying them under the influence of increased radiative forcing. It is shown that the neglected flux is particularly important in a decaying ice cover approaching the transitions to seasonal or ice-free conditions. Clearly, a mishandling of the evolution of the ice area has leading-order effects on the ice-albedo feedback. Accordingly, it may be of considerable importance to re-examine the relevant climate model schemes and to begin the process of converting them to fully resolve the sea ice thickness distribution in a manner such as remapping, which does not in principle suffer from the pathology we describe.

physics.ao-ph↗

arXiv2014

Surface tension and the mechanics of liquid inclusions in compliant solids

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and structures including the theories of composites, fracture, and plasticity. However, it does not include the effects of surface stress, which has recently been shown to control many processes in soft materials such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials, we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of the solid and the surface tension (\textit{i.e.} isotropic strain-independent surface stress) of the solid-liquid interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and far-field stress fields. These phenomenon depend strongly on the ratio of inclusion radius, $R$, to an elastocapillary length, $L$. Surface tension is significant whenever inclusions are smaller than $100L$. While Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our results show that liquid inclusions can actually stiffen a solid when $R<3L/2$. Intriguingly, surface tension cloaks the far-field signature of liquid inclusions when $R=3L/2$. These results are have far-reaching applications from measuring local stresses in biological tissue, to determining the failure strength of soft composites.

cond-mat.soft↗

arXiv2014

Stiffening solids with liquid inclusions

From bone and wood to concrete and carbon fibre, composites are ubiquitous natural and engineering materials. Eshelby's inclusion theory describes how macroscopic stress fields couple to isolated microscopic inclusions, allowing prediction of a composite's bulk mechanical properties from a knowledge of its microstructure. It has been extended to describe a wide variety of phenomena from solid fracture to cell adhesion. Here, we show experimentally and theoretically that Eshelby's theory breaks down for small liquid inclusions in a soft solid. In this limit, an isolated droplet's deformation is strongly size-dependent with the smallest droplets mimicking the behaviour of solid inclusions. Furthermore, in opposition to the predictions of conventional composite theory, we find that finite concentrations of small liquid inclusions enhance the stiffness of soft solids. A straight-forward extension of Eshelby's theory, accounting for the surface tension of the solid-liquid interface, explains our experimental observations. The counterintuitive effect of liquid-stiffening of solids is expected whenever droplet radii are smaller than an elastocapillary length, given by the ratio of the surface tension to Young's modulus of the solid matrix.

cond-mat.soft↗

arXiv2013

Hertz Beyond Belief

We examine the validity of Hertz's linear elastic theory for central collisions using a viscoelastic model. This model explains why Hertz's theory is accurate in predicting the collision time and maximum contact area even when $40\%$ of the kinetic energy is lost due to viscous dissipation. The main reason is that both the collision time and maximum contact area have a very weak dependence on the impact velocity. Moreover, we show that colliding objects exhibit an apparent size dependent yield strength, which results from larger objects dissipating less energy at a given impact velocity.

physics.class-ph↗

arXiv2013

Surface tension and contact with soft elastic solids

Johnson-Kendall-Robert (JKR) theory is the basis of modern contact mechanics. It describes how two deformable objects adhere together, driven by adhesion energy and opposed by elasticity. However, it does not include solid surface tension, which also opposes adhesion by acting to flatten the surface of soft solids. We tested JKR theory to see if solid surface tension affects indentation behaviour. Using confocal microscopy, we characterised the indentation of glass particles into soft, silicone substrates. While JKR theory held for particles larger than a critical, elastocapillary lengthscale, it failed for smaller particles. Instead, adhesion of small particles mimicked the adsorption of particles at a fluid interface, with a size-independent contact angle between the undeformed surface and the particle given by a generalised version of Young's law. A simple theory quantitatively captures this behaviour, and explains how solid surface tension dominates elasticity for small-scale indentation of soft materials.

cond-mat.soft↗

arXiv2013

Patterning droplets with durotaxis

Numerous cell-types have shown a remarkable ability to detect and move along gradients in stiffness of an underlying substrate -- a process known as durotaxis. The mechanisms underlying durotaxis are still unresolved, but generally believed to involve active sensing and locomotion. Here, we show that simple liquid droplets also undergo durotaxis. By modulating substrate stiffness, we obtain fine control of droplet position on soft, flat substrates. Unlike other control mechanisms, droplet durotaxis works without imposing chemical, thermal, electrical or topographical gradients. This enables a new approach to large-scale droplet patterning and is potentially useful for many applications, such as microfluidics, thermal control and microfabrication.

cond-mat.soft↗