Searcharxiv⌕ Search

arXiv subjects

J. S. Wettlaufer

Publications and source records attributed to J. S. Wettlaufer.

At least 37 records · Page 2Linked to original sources
arXiv2020

Analytical Survival Analysis of the Ornstein-Uhlenbeck Process

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein-Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in boundary layers, near the lateral boundaries of the domain. The analytic solution agrees extremely well with the numerical solution and takes into account the non-negligible leakage of probability that occurs at short times when the stochastic process begins close to one of the boundaries. Given the range of applications of Ornstein-Uhlenbeck processes, the analytic solution is of broad relevance across many fields of natural and engineering science.

cond-mat.stat-mech↗
arXiv2020

Transient Convective Spin-up Dynamics

We study the formation, longevity and breakdown of convective rings during impulsive spin-up in square and cylindrical containers using direct numerical simulations. The rings, which are axisymmetric alternating regions of up- and down-welling flow that can last for O (100) rotation times, were first demonstrated experimentally and arise due to a balance between Coriolis and viscous effects. We study the formation of these rings in the context of the Greenspan-Howard spin-up process, the disruption of which modifies ring formation and evolution. We show that, unless imprinted by boundary geometry, convective rings can only form when the surface providing buoyancy forcing is a free-slip surface, thereby explaining an apparent disagreement between experimental results in the literature. For Prandtl numbers from 1--5 we find that the longest-lived rings occur for intermediate Prandtl numbers, with a Rossby number dependence. Finally, we find that the constant evaporative heat-flux conditions imposed in the experiments are essential in sustaining the rings and in maintaining the vortices that form in consequence of the ring breakdown.

physics.flu-dyn↗
arXiv2020

Precursors to Rare Events in Stochastic Resonance

In stochastic resonance, a periodically forced Brownian particle in a double-well potential jumps between minima at rare increments, the prediction of which poses a major theoretical challenge. Here, we use a path-integral method to find a precursor to these transitions by determining the most probable (or "{optimal}") space-time path of a particle. We characterize the optimal path using a direct comparison principle between the Langevin and Hamiltonian dynamical descriptions, allowing us to express the jump condition in terms of the accumulation of noise around the stable periodic path. In consequence, as a system approaches a rare event these fluctuations approach one of the deterministic minimizers, thereby providing a precursor for predicting a stochastic transition. We demonstrate the method numerically, which allows us to determine whether a state is following a stable periodic path or will experience an incipient jump with a high probability. The vast range of systems that exhibit stochastic resonance behavior insures broad relevance of our framework, which allows one to extract precursor fluctuations from data.

physics.data-an↗
arXiv2019

Nonadiabatic escape and stochastic resonance

We analyze the fluctuation-driven escape of particles from a metastable state under the influence of a weak periodic force. We develop an asymptotic method to solve the appropriate Fokker-Planck equation with mixed natural and absorbing boundary conditions. The approach uses two boundary layers flanking an interior region; most of the probability is concentrated within the boundary layer near the metastable point of the potential and particles transit the interior region before exiting the domain through the other boundary layer, which is near the unstable maximal point of the potential. The dominant processes in each region are given by approximate time-dependent solutions matched to construct the approximate composite solution, which gives the rate of escape with weak periodic forcing. Using reflection we extend the method to a double well potential influenced by white noise and weak periodic forcing, and thereby derive a two-state stochastic model--the simplest treatment of stochastic resonance theory--in the nonadiabatic limit

cond-mat.stat-mech↗
arXiv2018

An intrinsic pink-noise multi-decadal global climate dynamics mode

Understanding multi-decadal variability is an essential goal of climate dynamics. For example, the recent phenomenon referred to as the "global warming hiatus" may reflect a coupling to an intrinsic, pre-industrial, multi-decadal variability process. Here, using a multi-fractal time series method, we demonstrate that forty-two data sets of seventy-nine proxies with global coverage exhibit pink noise characteristics on multi-decadal time scales. To quantify the persistence of this behavior, we examine high-resolution ice core and speleothem data to find pink noise in both pre- and post-industrial periods. We examine the spatial structure with Empirical Orthogonal Function (EOF) analysis of the monthly-averaged surface temperature from 1901 to 2012. The first mode clearly shows the distribution of ocean heat flux sinks located in the eastern Pacific and the Southern Ocean, and has pink noise characteristics on a multi-decadal time-scale. We hypothesize that this pink noise multi-decadal spatial mode may resonate with externally-driven greenhouse gas forcing, driving large-scale climate processes.

physics.ao-ph↗
arXiv2018

Penetrative Convection at High Rayleigh Numbers

We study penetrative convection of a fluid confined between two horizontal plates, the temperatures of which are such that a temperature of maximum density lies between them. The range of Rayleigh numbers studied is $Ra = \left[10^6, 10^8 \right]$ and the Prandtl numbers are $Pr = 1$ and $11.6$. An evolution equation for the growth of the convecting region is obtained through an integral energy balance. We identify a new non-dimensional parameter, $Λ$, which is the ratio of temperature difference between the stable and unstable regions of the flow; larger values of $Λ$ denote increased stability of the upper stable layer. We study the effects of $Λ$ on the flow field using well-resolved lattice Boltzmann simulations, and show that the characteristics of the flow depend sensitively upon it. For the range $Λ= \left[0.01, 4\right]$, we find that for a fixed $Ra$ the Nusselt number, $Nu$, increases with decreasing $Λ$. We also investigate the effects of $Λ$ on the vertical variation of convective heat flux and the Brunt-Väisälä frequency. Our results clearly indicate that in the limit $Λ\rightarrow 0$ the problem reduces to that of the classical Rayleigh-Bénard convection.

physics.flu-dyn↗
arXiv2015

Theory of the sea ice thickness distribution

We use concepts from statistical physics to transform the original evolution equation for the sea ice thickness distribution $g(h)$ due to Thorndike et al., (1975) into a Fokker-Planck like conservation law. The steady solution is $g(h) = {\cal N}(q) h^q \mathrm{e}^{-~ h/H}$, where $q$ and $H$ are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for $h \ll 1$, $g(h)$ is controlled by both thermodynamics and mechanics, whereas for $h \gg 1$ only mechanics controls $g(h)$. Finally, we derive the underlying Langevin equation governing the dynamics of the ice thickness $h$, from which we predict the observed $g(h)$. The genericity of our approach provides a framework for studying the geophysical scale structure of the ice pack using methods of broad relevance in statistical mechanics.

physics.ao-ph↗
arXiv2015

Maximal Stochastic Transport in the Lorenz Equations

We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of \citet{Doering15}, but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.

nlin.CD↗
arXiv2014

On the Interpretation of Stratonovich Calculus

The Ito-Stratonovich dilemma is revisited from the perspective of the interpretation of Stratonovich calculus using shot noise. Over the long time scales of the displacement of an observable, the principal issue is how to deal with finite/zero autocorrelation of the stochastic noise. The former (non-zero) noise autocorrelation structure preserves the normal chain rule using a mid-point selection scheme, which is the basis Stratonovich calculus, whereas the instantaneous autocorrelation structure of Ito's approach does not. By considering the finite decay of the noise correlations on time scales very short relative to the overall displacement times of the observable, we suggest a generalization of the integral Taylor expansion criterion of Wong and Zakai (1965) for the validity of the Stratonovich approach.

cond-mat.stat-mech↗
arXiv2013

Can planetesimals form by collisional fusion?

As a test bed for the growth of protoplanetary bodies in a turbulent circumstellar disk we examine the fate of a boulder using direct numerical simulations of particle seeded gas flowing around it. We provide an accurate description of the flow by imposing no-slip and non-penetrating boundary conditions on the boulder surface using the immersed boundary method pioneered by Peskin (2002). Advected by the turbulent disk flow, the dust grains collide with the boulder and we compute the probability density function (PDF) of the normal component of the collisional velocity. Through this examination of the statistics of collisional velocities we test the recently developed concept of collisional fusion which provides a physical basis for a range of collisional velocities exhibiting perfect sticking. A boulder can then grow sufficiently rapidly to settle into a Keplerian orbit on disk evolution time scales.

astro-ph.EP↗
arXiv2013

Grain boundary melting in ice

We describe an optical scattering study of grain boundary premelting in water ice. Ubiquitous long ranged attractive polarization forces act to suppress grain boundary melting whereas repulsive forces originating in screened Coulomb interactions and classical colligative effects enhance it. The liquid enhancing effects can be manipulated by adding dopant ions to the system. For all measured grain boundaries this leads to increasing premelted film thickness with increasing electrolyte concentration. Although we understand that the interfacial surface charge densities $q_s$ and solute concentrations can potentially dominate the film thickness, we can not directly measure them within a given grain boundary. Therefore, as a framework for interpreting the data we consider two appropriate $q_s$ dependent limits; one is dominated by the colligative effect and one is dominated by electrostatic interactions.

cond-mat.mtrl-sci↗
arXiv2012

Modeling the Influence of Antifreeze Proteins on Three-Dimensional Ice Crystal Melt Shapes using a Geometric Approach

The melting of pure axisymmetric ice crystals has been described previously by us within the framework of so-called geometric crystal growth. Nonequilibrium ice crystal shapes evolving in the presence of hyperactive antifreeze proteins (hypAFPs) are experimentally observed to assume ellipsoidal geometries ("lemon" or "rice" shapes). To analyze such shapes we harness the underlying symmetry of hexagonal ice Ih and extend two-dimensional geometric models to three-dimensions to reproduce the experimental dissolution process. The geometrical model developed will be useful as a quantitative test of the mechanisms of interaction between hypAFPs and ice.

physics.bio-ph↗
arXiv2012

On the existence of stable seasonally varying Arctic sea ice in simple models

Within the framework of lower order thermodynamic theories for the climatic evolution of Arctic sea ice we isolate the conditions required for the existence of stable seasonally-varying solutions, in which ice forms each winter and melts away each summer. This is done by constructing a two-season model from the continuously evolving theory of Eisenman and Wettlaufer (2009) and showing that seasonally-varying states are unstable under constant annual average short-wave radiative forcing. However, dividing the summer season into two intervals (ice covered and ice free) provides sufficient freedom to stabilize seasonal ice. Simple perturbation theory shows that the condition for stability is determined by when the ice vanishes in summer and hence the relative magnitudes of the summer heat flux over the ocean versus over the ice. This scenario is examined within the context of greenhouse gas warming, as a function of which stability conditions are discerned.

physics.ao-ph↗
arXiv2012

Trends, noise and reentrant long-term persistence in Arctic sea ice

We examine the long-term correlations and multifractal properties of daily satellite retrievals of Arctic sea ice albedo and extent, for periods of $\sim$ 23 years and 32 years respectively. The approach harnesses a recent development called Multifractal Temporally Weighted Detrended Fluctuation Analysis (MF-TWDFA), which exploits the intuition that points closer in time are more likely to be related than distant points. In both data sets we extract multiple crossover times, as characterized by generalized Hurst exponents, ranging from synoptic to decadal. The method goes beyond treatments that assume a single decay scale process, such as a first-order autoregression, which cannot be justifiably fit to these observations. Importantly, the strength of the seasonal cycle "masks" long term correlations on time scales beyond seasonal. When removing the seasonal cycle from the original record, the ice extent data exhibits white noise behavior from seasonal to bi-seasonal time scales, whereas the clear fingerprints of the short (weather) and long ($\sim$ 7 and 9 year) time scales remain, the latter associated with the recent decay in the ice cover. Therefore, long term persistence is reentrant beyond the seasonal scale and it is not possible to distinguish whether a given ice extent minimum/maximum will be followed by a minimum/maximum that is larger or smaller in magnitude.

physics.ao-ph↗
arXiv2011

Density functional theory for Baxter's sticky hard spheres in confinement

It has recently been shown that a free energy for Baxter's sticky hard sphere fluid is uniquely defined within the framework of fundamental measure theory (FMT) for the inhomogeneous hard sphere fluid, provided that it obeys scaled-particle theory and the Percus-Yevick (PY) result for the direct correlation function [Hansen-Goos and Wettlaufer, J. Chem. Phys. {\bf 134}, 014506 (2011)]. Here, combining weighted densities from common versions of FMT with a new vectorial weighted density, we derive a regularization of the divergences of the associated strongly confined limit. Moreover, the simple free energy that emerges is exact in the zero-dimensional limit, leaves the underlying equation of state unaffected, and yields a direct correlation function distinct from the PY expression. Comparison with simulation data for both the bulk pair correlation function and the density profiles in confinement shows that the new theory is significantly more accurate than the PY-based results. Finally, the resulting free energy is applicable to a glass of adhesive hard spheres.

cond-mat.soft↗
arXiv2011

A Low Order Theory of Arctic Sea Ice Stability

We analyze the stability of a low-order coupled sea ice and climate model and extract the essential physics governing the time scales of response as a function of greenhouse gas forcing. Under present climate conditions the stability is controlled by longwave radiation driven heat conduction. However, as greenhouse gas forcing increases and the ice cover decays, the destabilizing influence of ice-albedo feedback acts on equal footing with longwave stabilization. Both are seasonally out of phase and as the system warms towards a seasonal ice state these effects, which underlie the bifurcations between climate states, combine exhibiting a "slowing down" to extend the intrinsic relaxation time scale from ~ 2 yr to 5 yr.

physics.ao-ph↗
arXiv2011

Decadal to seasonal variability of Arctic sea ice albedo

A controlling factor in the seasonal and climatological evolution of the sea ice cover is its albedo $α$. Here we analyze Arctic data from the Advanced Very High Resolution Radiometer (AVHRR) Polar Pathfinder and assess the seasonality and variability of broadband albedo from a 23 year daily record. We produce a histogram of daily albedo over ice covered regions in which the principal albedo transitions are seen; high albedo in late winter and spring, the onset of snow melt and melt pond formation in the summer, and fall freeze up. The bimodal late summer distribution demonstrates the combination of the poleward progression of the onset of melt with the coexistence of perennial bare ice with melt ponds and open water, which then merge to a broad peak at $α\gtrsim $ 0.5. We find the interannual variability to be dominated by the low end of the $α$ distribution, highlighting the controlling influence of the ice thickness distribution and large-scale ice edge dynamics. The statistics obtained provide a simple framework for model studies of albedo parameterizations and sensitivities.

physics.ao-ph↗