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Academic Commons Search Resultsen-usMultiple events on single molecules: unbiased estimation in single-molecule biophysics.
http://academiccommons.columbia.edu/catalog/ac:177525
Koster, Daniel A.; Wiggins, Chris H.; Dekker, Nynke H.http://dx.doi.org/10.7916/D8HQ3XF2Sat, 20 Sep 2014 00:00:00 +0000Most analyses of single-molecule experiments consist of binning experimental outcomes into a histogram and finding the parameters that optimize the fit of this histogram to a given data model. Here we show that such an approach can introduce biases in the estimation of the parameters, thus great care must be taken in the estimation of model parameters from the experimental data. The bias can be particularly large when the observations themselves are not statistically independent and are subjected to global constraints, as, for example, when the iterated steps of a motor protein acting on a single molecule must not exceed the total molecule length. We have developed a maximum-likelihood analysis, respecting the experimental constraints, which allows for a robust and unbiased estimation of the parameters, even when the bias well exceeds 100%. We demonstrate the potential of the method for a number of single-molecule experiments, focusing on the removal of DNA supercoils by topoisomerase IB, and validate the method by numerical simulation of the experiment.Biophysics, Applied mathematicschw2Applied Physics and Applied MathematicsArticlesInferring network mechanisms: The Drosophila melanogaster protein interaction network
http://academiccommons.columbia.edu/catalog/ac:177528
Middendorf, Manuel; Ziv, Etay; Wiggins, Chris H.http://dx.doi.org/10.7916/D8862DZVSat, 20 Sep 2014 00:00:00 +0000Naturally occurring networks exhibit quantitative features revealing underlying growth mechanisms. Numerous network mechanisms have recently been proposed to reproduce specific properties such as degree distributions or clustering coefficients. We present a method for inferring the mechanism most accurately capturing a given network topology, exploiting discriminative tools from machine learning. The Drosophila melanogaster protein network is confidently and robustly (to noise and training data subsampling) classified as a duplication–mutation–complementation network over preferential attachment, small-world, and a duplication–mutation mechanism without complementation. Systematic classification, rather than statistical study of specific properties, provides a discriminative approach to understand the design of complex networks.Applied mathematicschw2Applied Physics and Applied MathematicsArticlesA stochastic spectral analysis of transcriptional regulatory cascades
http://academiccommons.columbia.edu/catalog/ac:177519
Walczak, Aleksandra M.; Mugler, Andrew; Wiggins, Chris H.http://dx.doi.org/10.7916/D81N7ZNNSat, 20 Sep 2014 00:00:00 +0000The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here, we introduce the spectral method for computation of the joint probability distribution over all species in a biological network. The spectral method exploits the natural eigenfunctions of the master equation of birth – death processes to solve for the joint distribution of modules within the network, which then inform each other and facilitate calculation of the entire joint distribution. We illustrate the method on a ubiquitous case in nature: linear regulatory cascades. The efficiency of the method makes possible numerical optimization of the input and regulatory parameters, revealing design properties of, e.g., the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of up-regulations outperforms a chain of down-regulations. We anticipate that this numerical approach may be useful for modeling noise in a variety of small network topologies in biology.Applied mathematicschw2Applied Physics and Applied MathematicsArticlesFast dynamics of supercoiled DNA revealed by single-molecule experiments.
http://academiccommons.columbia.edu/catalog/ac:177522
Crut, Aurelien; Koster, Daniel A.; Seidel, Ralf; Wiggins, Chris H.; Dekker, Nynke H.http://dx.doi.org/10.7916/D8S75DW3Sat, 20 Sep 2014 00:00:00 +0000The dynamics of supercoiled DNA play an important role in various cellular processes such as transcription and replication that involve DNA supercoiling. We present experiments that enhance our understanding of these dynamics by measuring the intrinsic response of single DNA molecules to sudden changes in tension or torsion. The observed dynamics can be accurately described by quasistatic models, independent of the degree of supercoiling initially present in the molecules. In particular, the dynamics are not affected by the continuous removal of the plectonemes. These results set an upper bound on the hydrodynamic drag opposing plectoneme removal, and thus provide a quantitative baseline for the dynamics of bare DNA.Biophysics, Biomechanicschw2Applied Physics and Applied MathematicsArticlesTrapping and Wiggling: Elastohydrodynamics of Driven Microfilaments
http://academiccommons.columbia.edu/catalog/ac:177420
Wiggins, Chris H.; Riveline, D.; Ott, A.; Goldstein, Raymond E.http://dx.doi.org/10.7916/D8QJ7FTVFri, 19 Sep 2014 00:00:00 +0000We present an analysis of the planar motion of single semiflexible filaments subject to viscous drag or point forcing. These are the relevant forces in dynamic experiments designed to measure biopolymer bending moduli. By analogy with the “Stokes problems” in hydrodynamics (motion of a viscous fluid induced by that of a wall bounding the fluid), we consider the motion of a polymer, one end of which is moved in an impulsive or oscillatory way. Analytical solutions for the time-dependent shapes of such moving polymers are obtained within an analysis applicable to small-amplitude deformations. In the case of oscillatory driving, particular attention is paid to a characteristic length determined by the frequency of oscillation, the polymer persistence length, and the viscous drag coefficient. Experiments on actin filaments manipulated with optical traps confirm the scaling law predicted by the analysis and provide a new technique for measuring the elastic bending modulus. Exploiting this model, we also present a reanalysis of several published experiments on microtubules.Biophysicschw2Applied Physics and Applied MathematicsArticlesA classification-based framework for predicting and analyzing gene regulatory response
http://academiccommons.columbia.edu/catalog/ac:177435
Kundaje, Anshul; Middendorf, Manuel; Wiggins, Chris; Shah, Mihir; Freund, Yoav; Leslie, Christinahttp://dx.doi.org/10.7916/D82V2DNHFri, 19 Sep 2014 00:00:00 +0000We have recently introduced a predictive framework for studying gene transcriptional regulation in simpler organisms using a novel supervised learning algorithm called GeneClass. GeneClass is motivated by the hypothesis that in model organisms such as Saccharomyces cerevisiae, we can learn a decision rule for predicting whether a gene is up- or down-regulated in a particular microarray experiment based on the presence of binding site subsequences ("motifs") in the gene's regulatory region and the expression levels of regulators such as transcription factors in the experiment ("parents"). GeneClass formulates the learning task as a classification problem — predicting +1 and -1 labels corresponding to up- and down-regulation beyond the levels of biological and measurement noise in microarray measurements. Using the Adaboost algorithm, GeneClass learns a prediction function in the form of an alternating decision tree, a margin-based generalization of a decision tree. In the current work, we introduce a new, robust version of the GeneClass algorithm that increases stability and computational efficiency, yielding a more scalable and reliable predictive model. The improved stability of the prediction tree enables us to introduce a detailed post-processing framework for biological interpretation, including individual and group target gene analysis to reveal condition-specific regulation programs and to suggest signaling pathways. Robust GeneClass uses a novel stabilized variant of boosting that allows a set of correlated features, rather than single features, to be included at nodes of the tree; in this way, biologically important features that are correlated with the single best feature are retained rather than decorrelated and lost in the next round of boosting. Other computational developments include fast matrix computation of the loss function for all features, allowing scalability to large datasets, and the use of abstaining weak rules, which results in a more shallow and interpretable tree. We also show how to incorporate genome-wide protein-DNA binding data from ChIP chip experiments into the GeneClass algorithm, and we use an improved noise model for gene expression data. Using the improved scalability of Robust GeneClass, we present larger scale experiments on a yeast environmental stress dataset, training and testing on all genes and using a comprehensive set of potential regulators. We demonstrate the improved stability of the features in the learned prediction tree, and we show the utility of the post-processing framework by analyzing two groups of genes in yeast — the protein chaperones and a set of putative targets of the Nrg1 and Nrg2 transcription factors — and suggesting novel hypotheses about their transcriptional and post-transcriptional regulation. Detailed results and Robust GeneClass source code is available for download from http://www.cs.columbia.edu/compbio/robust-geneclassBioinformatics, Geneticschw2Applied Physics and Applied MathematicsArticlesDynamics of semiflexible polymers in a flow field
http://academiccommons.columbia.edu/catalog/ac:177471
Munk, Tobias; Hallatschek, Oskar; Wiggins, Chris H.; Frey, Erwinhttp://dx.doi.org/10.7916/D85M647VFri, 19 Sep 2014 00:00:00 +0000We present a method to investigate the dynamics of a single semiflexible polymer, subject to anisotropic friction in a viscous fluid. In contrast to previous approaches, we do not rely on a discrete bead-rod model, but introduce a suitable normal mode decomposition of a continuous space curve. By means of a perturbation expansion for stiff filaments, we derive a closed set of coupled Langevin equations in mode space for the nonlinear dynamics in two dimensions, taking into account exactly the local constraint of inextensibility. The stochastic differential equations obtained this way are solved numerically, with parameters adjusted to describe the motion of actin filaments. As an example, we show results for the tumbling motion in shear flow.Applied mathematics, Bioinformaticschw2Applied Physics and Applied MathematicsArticlesInformation-theoretic approach to network modularity
http://academiccommons.columbia.edu/catalog/ac:177474
Ziv, Etay; Middendorf, Manuel; Wiggins, Chris H.http://dx.doi.org/10.7916/D8X34VZVFri, 19 Sep 2014 00:00:00 +0000Exploiting recent developments in information theory, we propose, illustrate, and validate a principled information-theoretic algorithm for module discovery and the resulting measure of network modularity. This measure is an order parameter (a dimensionless number between 0 and 1). Comparison is made with other approaches to module discovery and to quantifying network modularity (using Monte Carlo generated Erdös-like modular networks). Finally, the network information bottleneck (NIB) algorithm is applied to a number of real world networks, including the “social” network of coauthors at the 2004 APS March Meeting.Applied mathematicschw2Applied Physics and Applied MathematicsArticlesSystematic identification of statistically significant network measures
http://academiccommons.columbia.edu/catalog/ac:177477
Ziv, Etay; Koytcheff, Robin; Middendorf, Manuel; Wiggins, Chris H.http://dx.doi.org/10.7916/D8NK3CJNFri, 19 Sep 2014 00:00:00 +0000We present a graph embedding space (i.e., a set of measures on graphs) for performing statistical analyses of networks. Key improvements over existing approaches include discovery of “motif hubs” (multiple overlapping significant subgraphs), computational efficiency relative to subgraph census, and flexibility (the method is easily generalizable to weighted and signed graphs). The embedding space is based on scalars, functionals of the adjacency matrix representing the network. Scalars are global, involving all nodes; although they can be related to subgraph enumeration, there is not a one-to-one mapping between scalars and subgraphs. Improvements in network randomization and significance testing—we learn the distribution rather than assuming Gaussianity—are also presented. The resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine-learning techniques for network classification.Applied mathematicschw2Applied Physics and Applied MathematicsArticlesSpectral solutions to stochastic models of gene expression with bursts and regulation
http://academiccommons.columbia.edu/catalog/ac:177467
Mugler, Andrew; Walczak, Aleksandra M.; Wiggins, Chris H.http://dx.doi.org/10.7916/D8F18X86Fri, 19 Sep 2014 00:00:00 +0000Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.Applied mathematics, Bioinformaticschw2Applied Physics and Applied MathematicsArticlesTime-dependent information transmission in a model regulatory circuit
http://academiccommons.columbia.edu/catalog/ac:177464
, F. Mancini; Wiggins, Chris H.; Marsili, M.; Walczak, Aleksandra. M.http://dx.doi.org/10.7916/D8PK0DNKFri, 19 Sep 2014 00:00:00 +0000Many biological regulatory systems respond with a physiological delay when processing signals. A simple model of regulation which respects these features shows how the ability of a delayed output to transmit information is limited: at short times by the time scale of the dynamic input, at long times by that of the dynamic output. We find that topologies of maximally informative networks correspond to commonly occurring biological circuits linked to stress response and that circuits functioning out of steady state may exploit absorbing states to transmit information optimally.Applied mathematics, Bioinformaticschw2Applied Physics and Applied MathematicsArticlesViscous Nonlinear Dynamics of Twist and Writhe
http://academiccommons.columbia.edu/catalog/ac:177498
Goldstein, Raymond E.; Powers, Thomas R.; Wiggins, Chris H.http://dx.doi.org/10.7916/D8V40SRJFri, 19 Sep 2014 00:00:00 +0000Exploiting the “natural” frame of space curves, we formulate an intrinsic dynamics of a twisted elastic filament in a viscous fluid. Coupled nonlinear equations describing the temporal evolution of the filament's complex curvature and twist density capture the dynamic interplay of twist and writhe. These equations are used to illustrate a remarkable nonlinear phenomenon: geometric untwisting of open filaments, whereby twisting strains relax through a transient writhing instability without axial rotation. Experimentally observed writhing motions of fibers of the bacterium B. subtilis [N. H. Mendelson et al., J. Bacteriol. 177, 7060 (1995)] may be examples of this untwisting process.Biophysicschw2Applied Physics and Applied MathematicsArticlesFlexive and Propulsive Dynamics of Elastica at Low Reynolds Number
http://academiccommons.columbia.edu/catalog/ac:177501
Wiggins, Chris H.; Goldstein, Raymond E.http://dx.doi.org/10.7916/D8KP80P6Fri, 19 Sep 2014 00:00:00 +0000A stiff one-armed swimmer in glycerine goes nowhere. However, if its arm is elastic, the swimmer can go on its way. Quantifying this observation, we study a hyperdiffusion equation for the shape of the elastica in a viscous fluid, find solutions for impulsive or oscillatory forcing, and elucidate relevant aspects of propulsion. These results have application in a variety of physical and biological contexts, from dynamic experiments measuring biopolymer bending moduli to instabilities of twisted elastic filaments.Biophysics, Applied mathematicschw2Applied Physics and Applied MathematicsArticlesBayesian Approach to Network Modularity
http://academiccommons.columbia.edu/catalog/ac:177486
Mugler, Andrew; Wiggins, Chris H.http://dx.doi.org/10.7916/D8W37TTGFri, 19 Sep 2014 00:00:00 +0000We present an efficient, principled, and interpretable technique for inferring module assignments and for identifying the optimal number of modules in a given network. We show how several existing methods for finding modules can be described as variant, special, or limiting cases of our work, and how the method overcomes the resolution limit problem, accurately recovering the true number of modules. Our approach is based on Bayesian methods for model selection which have been used with success for almost a century, implemented using a variational technique developed only in the past decade. We apply the technique to synthetic and real networks and outline how the method naturally allows selection among competing models.Biophysics, Applied mathematicschw2Applied Physics and Applied MathematicsArticlesInformation-Optimal Transcriptional Response to Oscillatory Driving
http://academiccommons.columbia.edu/catalog/ac:177483
Mugler, Andrew; Walczak, Aleksandra M.; Wiggins, Chris H.http://dx.doi.org/10.7916/D84M9321Fri, 19 Sep 2014 00:00:00 +0000Intracellular transmission of information via chemical and transcriptional networks is thwarted by a physical limitation: The finite copy number of the constituent chemical species introduces unavoidable intrinsic noise. Here we solve for the complete probabilistic description of the intrinsically noisy response to an oscillatory driving signal. We derive and numerically verify a number of simple scaling laws. Unlike in the case of measuring a static quantity, response to an oscillatory signal can exhibit a resonant frequency which maximizes information transmission. Furthermore, we show that the optimal regulatory design is dependent on biophysical constraints (i.e., the allowed copy number and response time). The resulting phase diagram illustrates under what conditions threshold regulation outperforms linear regulation.Biophysics, Applied mathematicschw2Applied Physics and Applied MathematicsArticlesRelaxation Dynamics of Semiflexible Polymers
http://academiccommons.columbia.edu/catalog/ac:177492
Bohbot-Raviv, Y.; Zhao, W. Z.; Feingold, M.; Wiggins, Chris H.; Granek, R.http://dx.doi.org/10.7916/D8C53JD9Fri, 19 Sep 2014 00:00:00 +0000We study the relaxation dynamics of a semiflexible chain by introducing a time-dependent tension. The chain has one of its ends attached to a large bead, and the other end is fixed. We focus on the initial relaxation of the chain that is initially strongly stretched. Using a tension that is self-consistently determined, we obtain the evolution of the end-to-end distance with no free parameters. Our results are in good agreement with single molecule experiments on double stranded DNA.Biophysicschw2Applied Physics and Applied MathematicsArticlesElastohydrodynamic study of actin filaments using fluorescence microscopy
http://academiccommons.columbia.edu/catalog/ac:177480
Riveline, D.; Wiggins, Chris H.; Golstein, Raymond E.; Ott, A.http://dx.doi.org/10.7916/D8D50KGSFri, 19 Sep 2014 00:00:00 +0000We probed the bending of actin subject to external forcing and viscous drag. Single actin filaments were moved perpendicular to their long axis in an oscillatory way by means of an optically tweezed latex bead attached to one end of the filaments. Shapes of these polymers were observed by epifluorescence microscopy. They were found to be in agreement with predictions of semiflexible polymer theory and slender-body hydrodynamics. A persistence length of 7.4±0.2 μm could be extracted.Biophysicschw2Applied Physics and Applied MathematicsArticlesLearning ‘‘graph-mer’’ Motifs that Predict Gene Expression Trajectories in Development
http://academiccommons.columbia.edu/catalog/ac:177504
Li, Xuejing; Panea, Casandra; Wiggins, Chris; Reinke, Valerie; Leslie, Christinahttp://dx.doi.org/10.7916/D8B56H8XFri, 19 Sep 2014 00:00:00 +0000A key problem in understanding transcriptional regulatory networks is deciphering what cis regulatory logic is encoded in gene promoter sequences and how this sequence information maps to expression. A typical computational approach to this problem involves clustering genes by their expression profiles and then searching for overrepresented motifs in the promoter sequences of genes in a cluster. However, genes with similar expression profiles may be controlled by distinct regulatory programs. Moreover, if many gene expression profiles in a data set are highly correlated, as in the case of whole organism developmental time series, it may be difficult to resolve fine-grained clusters in the first place. We present a predictive framework for modeling the natural flow of information, from promoter sequence to expression, to learn cis regulatory motifs and characterize gene expression patterns in developmental time courses. We introduce a cluster-free algorithm based on a graph-regularized version of partial least squares (PLS) regression to learn sequence patterns—represented by graphs of k-mers, or “graph-mers”—that predict gene expression trajectories. Applying the approach to wildtype germline development in Caenorhabditis elegans, we found that the first and second latent PLS factors mapped to expression profiles for oocyte and sperm genes, respectively. We extracted both known and novel motifs from the graph-mers associated to these germline-specific patterns, including novel CG-rich motifs specific to oocyte genes. We found evidence supporting the functional relevance of these putative regulatory elements through analysis of positional bias, motif conservation and in situ gene expression. This study demonstrates that our regression model can learn biologically meaningful latent structure and identify potentially functional motifs from subtle developmental time course expression data.Applied mathematics, Geneticschw2Applied Physics and Applied MathematicsArticlesDynamic Patterns and Self-Knotting of a Driven Hanging Chain
http://academiccommons.columbia.edu/catalog/ac:177495
Belmonte, Andrew; Shelley, Michael J.; Eldakar, Shaden T.; Wiggins, Chris H.http://dx.doi.org/10.7916/D83N21XSFri, 19 Sep 2014 00:00:00 +0000When shaken vertically, a hanging chain displays a startling variety of distinct behaviors. We find experimentally that instabilities occur in tonguelike bands of parameter space, to swinging or rotating pendular motion, or to chaotic states. Mathematically, the dynamics are described by a nonlinear wave equation. A linear stability analysis predicts instabilities within the well-known resonance tongues; their boundaries agree very well with experiment. Full simulations of the 3D dynamics reproduce and elucidate many aspects of the experiment. The chain is also observed to tie knots in itself, some quite complex. This is beyond the reach of the current analysis and simulations.Biophysicschw2Applied Physics and Applied MathematicsArticlesWave and Vortex Dynamics on the Surface of a Sphere
http://academiccommons.columbia.edu/catalog/ac:167163
Polvani, Lorenzo M.; Dritschel, David G.http://dx.doi.org/10.7916/D8VD6WCWTue, 19 Nov 2013 00:00:00 +0000Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude. We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices. We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn’s ‘ribbon’.Atmospheric sciences, Applied mathematics, Aeronomylmp3Applied Physics and Applied MathematicsArticlesThe Roll-Up of Vorticity Strips on the Surface of a Sphere
http://academiccommons.columbia.edu/catalog/ac:167167
Dritschel, David G.; Polvani, Lorenzo M.http://dx.doi.org/10.7916/D8QN64NJTue, 19 Nov 2013 00:00:00 +0000We derive the conditions for the stability of strips or filaments of vorticity on the surface of a sphere. We find that the spherical results are surprisingly different from the planar ones, owing to the nature of the spherical geometry. Strips of vorticity on the surface of a sphere show a greater tendency to roll-up into vortices than do strips on a planar surface. The results are obtained by performing a linear stability analysis of the simplest, piecewise-constant vorticity configuration, namely a zonal band of uniform vorticity located in equilibrium between two latitudes. The presence of polar vortices is also considered, this having the effect of introducing adverse shear, a known stabilizing mechanism for planar flows. In several representative examples, the fully developed stages of the instabilities are illustrated by direct numerical simulation. The implication for planetary atmospheres is that barotropic flows on the sphere have a more pronounced tendency to produce small, long-lived vortices, especially in equatorial and mid-latitude regions, than was previously anticipated from the theoretical results for planar flows. Essentially, the curvature of the sphere's surface weakens the interaction between different parts of the flow, enabling these parts to behave in relative isolation.Atmospheric sciences, Applied mathematics, Aeronomylmp3Applied Physics and Applied MathematicsArticlesChaotic Lagrangian Trajectories around an Elliptical Vortex Patch Embedded in a Constant and Uniform Background Shear Flow
http://academiccommons.columbia.edu/catalog/ac:167176
Polvani, Lorenzo M.; Wisdom, J.http://dx.doi.org/10.7916/D8BG2KWDTue, 19 Nov 2013 00:00:00 +0000The Lagrangian flow around a Kida vortex [J. Phys. Soc. Jpn. 5 0, 3517 (1981)], an elliptical two‐dimensional vortex patch embedded in a uniform and constant background shear, is described by a nonintegrable two‐degree‐of‐freedom Hamiltonian. For small values of shear, there exist large chaotic zones surrounding the vortex, often much larger than the vortex itself and extremely close to its boundary. Motion within the vortex is integrable. Implications for two‐dimensional turbulence are discussed.Physics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesTwo-Layer Geostrophic Vortex Dynamics. Part 1. Upper-Layer V-States and Merger
http://academiccommons.columbia.edu/catalog/ac:167179
Polvani, Lorenzo M.; Flierl, G. R.; Zabusky, N. J.http://dx.doi.org/10.7916/D86Q1V53Tue, 19 Nov 2013 00:00:00 +0000We generalize the methods of two-dimensional contour dynamics to study a two-layer rotating fluid that obeys the quasi-geostrophic equations. We consider here only the case of a constant-potential-vorticity lower layer. We derive equilibrium solutions for monopolar (rotating) and dipolar (translating) geostrophic vortices in the upper layer, and compare them with the Euler case. We show that the equivalent barotropic (infinite lower layer) case is a singular limit of the two-layer system. We also investigate the effect of a finite lower layer on the merger of two regions of equal-sign potential vorticity in the upper layer. We discuss our results in the light of the recent laboratory experiments of Griffiths and Hopfinger (1986). The process of filamentation is found to be greatly suppressed for equivalent barotropic dynamics on scales larger than the radius of deformation. We show that the variation of the critical initial distance for merger as a function of the radius of deformation and the ratio of the layers at rest is closely related to the existence of vortex-pair equilibria and their geometrical properties.Applied mathematics, Physics, Atmospheric scienceslmp3Applied Physics and Applied MathematicsArticlesFilamentation of Unstable Vortex Structures via Separatrix Crossing: A Quantitative Estimate of Onset Time
http://academiccommons.columbia.edu/catalog/ac:167182
Polvani, Lorenzo M.; Flierl, G. R.; Zabusky, N. J.http://dx.doi.org/10.7916/D82Z13FRTue, 19 Nov 2013 00:00:00 +0000The onset of filamentation for compact vortex structures in two-dimensional incompressible flows is elucidated. An estimate is presented for the filamentation time of an unstably perturbed Kirchhoff ellipse, obtained from a linear analysis of the geometry of the instantaneous corotating streamfunction.Applied mathematics, Physicslmp3Applied Physics and Applied MathematicsArticlesRossby Wave Breaking, Microbreaking, Filamentation, and Secondary Vortex Formation: The Dynamics of a Perturbed Vortex
http://academiccommons.columbia.edu/catalog/ac:167170
Polvani, Lorenzo M.; Plumb, R. Alanhttp://dx.doi.org/10.7916/D8KW5CXTTue, 19 Nov 2013 00:00:00 +0000The behavior of an isolated vortex perturbed by topographically forced Rossby waves is studied using the method of Contour Dynamics. For a single-contour vortex a distinct forcing threshold exists above which the wave breaks in a dynamically significant way, leading to a disruption of the vortex. This breaking is distinguished from the process of weak filamentary breaking described by Dritschel and classified here as microbreaking; the latter occurs in nondivergent flow even at very small forcing amplitudes but does not affect the vortex in a substantial manner. In cases with finite Rossby deformation radius (comparable with the vortex radius) neither breaking nor microbreaking occurs below the forcing threshold. In common with previous studies using high-resolution spectral models, the vortex is not diluted by intrusion of outside air, except during remerger with a secondary vortex shed previously from the main vortex during a breaking event. The kinematics of the breaking process and of the vortex interior and the morphology of material ejected from the vortex are described. When the Rossby radius is finite there is substantial mixing in the deep interior of the vortex, even when the vortex is only mildly disturbed. Implications for the stratospheric polar vortex are discussed.Atmospheric sciences, Aeronomy, Physicslmp3Applied Physics and Applied MathematicsArticlesTwo-Layer Geostrophic Vortex Dynamics. Part 2. Alignment and Two-Layer V-States
http://academiccommons.columbia.edu/catalog/ac:167173
Lorenzo M., Polvanihttp://dx.doi.org/10.7916/D8G44N6TTue, 19 Nov 2013 00:00:00 +0000The process of alignment, a new fundamental interaction between vortices in a stratified and rapidly rotating fluid, is defined and studied in detail in the context of the two-layer quasi-geostrophic model. Alignment occurs when two vortices in different density layers coalesce by reducing their horizontal separation. It is found that only vortices whose radii are comparable with or larger than the Rossby deformation radius can align. In the same way as the merger process (in a single two dimensional layer) is related to the reverse energy cascade of two-dimensional turbulence, geostrophic potential vorticity alignment is related the barotropic-to baroclinic energy cascade of geostrophic turbulence in two layers. It is also shown how alignment is intimately connected with the existence of two-layer doubly connected geostrophic potential vorticity equilibria (V-states), for which the analysis of the geometry of the stream function in the corotating frame is found to be a crucial diagnostic. The finite-area analogues of the hetons of Hogg and Stommel (1985) are also determined : they consist of a propagating pair of opposite-signed potential vorticity patches located in different layers.Atmospheric sciences, , , Physics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesNonlinear, Barotropic Response to a Localized Topographic Forcing: Formation of a “Tropical Surf Zone” and Its Effect on Interhemispheric Propagation
http://academiccommons.columbia.edu/catalog/ac:167160
Waugh, D. W.; Plumb, R. A.; Polvani, Lorenzo M.http://dx.doi.org/10.7916/D8057CV5Tue, 19 Nov 2013 00:00:00 +0000The nonlinear response of a barotropic, nondivergent, spherical flow representative of the upper troposphere (but without a tropical Hadley cell) to localized, extratropical topographic forcing is examined using high-resolution contour surgery calculations. The response is shown to vary greatly with forcing amplitude and can be significantly different from the linear response. At large amplitude, Rossby wave breaking occurs in the tropics irrespective of the direction of the equatorial winds, and leads to small-scale stirring and the formation of a “tropical surf zone,” which inhibits the meridional propagation of the disturbance.Atmospheric sciences, Meteorology, Aeronomylmp3Applied Physics and Applied MathematicsArticlesProduction of Heavy Particles by Protons on Protons
http://academiccommons.columbia.edu/catalog/ac:167194
Afek, Y.; Margolis, B.; Polvani, Lorenzo M.http://dx.doi.org/10.7916/D8JW8BSFTue, 19 Nov 2013 00:00:00 +0000We calculate the production of heavy particles in the multi-GeV energy range using parton-model and statistical considerations. We discuss both central production and fragmentation. Our picture has implications for the question of the existence of a limiting temperature in hadron interactions.Physics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesThe Effect of Dissipation on Spatially Growing Nonlinear Baroclinic Waves
http://academiccommons.columbia.edu/catalog/ac:167185
Polvani, Lorenzo M.; Pedlosky, J.http://dx.doi.org/10.7916/D8Z60KZCTue, 19 Nov 2013 00:00:00 +0000The question of convective (i.e., spatial) instability of baroclinic waves on an f-plane is studied in the context of the two-layer model. The viscous and inviscid marginal curves for linear convective instability are obtained. The finite-amplitude problem shows that when dissipation is O(1) it acts to stabilize the waves that are of Eady type. For very small dissipation the weakly nonlinear analysis reveals that at low frequencies, contrary to what is known to occur in the temporal problem, in addition to the baroclinic component a barotropic correction to the “mean” flow is generated by the nonlinearities, and spatial equilibration occurs provided the ratio of shear to mean flow does not exceed some critical value. In the same limit, the slightly dissipative nonlinear dynamics reveals the presence of large spatial vacillations immediately downstream of the source, even if asymptotically (i.e., very far away from the source) the amplitudes are found to reach steady values. No case of period doubling or aperiodic behavior was found. The results obtained seem to be qualitatively independent of the form chosen to model the dissipation.Atmospheric sciences, Physics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesGeneralized Kirchhoff Vortices
http://academiccommons.columbia.edu/catalog/ac:167191
Polvani, Lorenzo M.; Flierl, G. R.http://dx.doi.org/10.7916/D8PN93H5Tue, 19 Nov 2013 00:00:00 +0000A family of exact solutions of the Euler equations is presented: they are generalizations of the Kirchhoff vortex to N confocal ellipses. Special attention is given to the case N=2, for which the stability is analyzed with a method similar to the one used by Love [Proc. London Math. Soc. 1, XXV 18 (1893)] for the Kirchhoff vortex. The results are compared with those for the corresponding circular problem.Applied mathematics, Physicslmp3Applied Physics and Applied MathematicsArticlesWave–Wave Interaction of Unstable Baroclinic Waves
http://academiccommons.columbia.edu/catalog/ac:167188
Pedlosky, Joseph; Polvani, Lorenzo M.http://dx.doi.org/10.7916/D8TD9V7HTue, 19 Nov 2013 00:00:00 +0000Two slightly unstable baroclinic waves in the two-layer Phillips model are allowed to interact with each other as well as the mean flow. A theory for small dissipation rates is developed to examine the role of wave–wave interaction in the dynamics of vacillation and aperiodicity in unstable systems. It is shown that the form of the dissipation mechanism as well as the overall dissipation timescale determines the nature of the dynamics. In particular, dissipation proportional to potential vorticity is shown to expunge amplitude vacillation due to wave–mean flow interactions. Wave–wave interaction, however, can yield amplitude vacillation. As the dissipation is decreased, the solutions evolve from steady waves (although propagating) to periodic vacillation until finally at small dissipation rates, chaotic behavior is obtained. This occurs in a range of relative growth rates of the two waves which depends on the strength of the wave–wave and wave–mean flow interactions.Atmospheric sciences, Applied mathematics, Physicslmp3Applied Physics and Applied MathematicsArticlesNonlinear Geostrophic Adjustment, Cyclone/Anticyclone Asymmetry, and Potential Vorticity Rearrangement
http://academiccommons.columbia.edu/catalog/ac:167047
Kuo, Allen C.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22162Fri, 08 Nov 2013 00:00:00 +0000Within the context of the rotating shallow water equations, it is shown how initially unbalanced states possessing certain symmetries dynamically evolve to lose those symmetries during nonlinear geostrophic adjustment. Using conservation law methods, it is demonstrated that the adjustment of equal and opposite (circular) mass imbalances results in a balanced end state where cyclones are stronger than anticyclones; the reverse holds true for momentum imbalances. In both cases, the degree of this asymmetry is shown to be directly proportional to the amount of initial imbalance (a measure of the nonlinearity occurring during time-dependent adjustment). On the other hand, the degree of asymmetry is maximal for imbalances of Rossby deformation scale. As for the potential vorticity, it is shown that its final profile can be noticeably different from its initial one; from an Eulerian perspective, this rearrangement is not confined to uniform shifts of potential vorticity fronts. Direct 2D numerical initial value problems confirm the asymmetry in the predicted final states and establish a relatively fast time scale for adjustment to complete. The robustness of these results is confirmed by studying, in addition, the adjustment of elliptical mass imbalances. The numerical integrations reveal that, during geostrophic adjustment, potential vorticity rearrangement occurs irreversibly on a fast wave time scale.Physics, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesWave-Vortex Interaction in Rotating Shallow Water. Part 1. One Space Dimension
http://academiccommons.columbia.edu/catalog/ac:167056
Kuo, Allen C.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22165Fri, 08 Nov 2013 00:00:00 +0000Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.Geophysics, Applied mathematics, Physicslmp3Applied Physics and Applied MathematicsArticlesBarotropic Vortex Pairs on a Rotating Sphere
http://academiccommons.columbia.edu/catalog/ac:167068
DiBattista, Mark T.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22169Fri, 08 Nov 2013 00:00:00 +0000Using a barotropic model in spherical geometry, we construct new solutions for steadily travelling vortex pairs and study their stability properties. We consider pairs composed of both point and finite-area vortices, and we represent the rotating background with a set of zonal strips of uniform vorticity. After constructing the solution for a single point-vortex pair, we embed it in a rotating background, and determine the equilibrium configurations that travel at constant speed without changing shape. For equilibrium solutions, we find that the stability depends on the relative strength (which may be positive or negative) of the vortex pair to the rotating background: eastward-travelling pairs are always stable, while westward-travelling pairs are unstable when their speeds approach that of the linear Rossby–Haurwitz waves. This finding also applies (with minor differences) to the case when the vortices are of finite area; in that case we find that, in addition to the point-vortex-like instabilities, the rotating background excites some finite-area instabilities for vortex pairs that would otherwise be stable. As for practical applications to blocking events, for which the slow westward pairs are relevant, our results indicate that free barotropic solutions are highly unstable, and thus suggest that forcing mechanisms must play an important role in maintaining atmospheric blocking events.Atmospheric sciences, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesThe Emergence of Jets and Vortices in Freely Evolving, Shallow‐Water Turbulence on a Sphere
http://academiccommons.columbia.edu/catalog/ac:167074
Cho, James Y‐K.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22171Fri, 08 Nov 2013 00:00:00 +0000Results from a series of simulations of unforced turbulence evolving within a shallow layer of fluid on a rotating sphere are presented. Simulations show that the turbulent evolution in the spherical domain is strongly dependent on numerical and physical conditions. The independent effects of (1) (hyper)dissipation and initial spectrum, (2) rotation rate, and (3) Rossby deformation radius are carefully isolated and studied in detail. In the nondivergent and nonrotating case, an initially turbulent flow evolves into a vorticityquadrupole at long times, a direct consequence of angular momentumconservation. In the presence of sufficiently strong rotation, the nondivergent long‐time behavior yields a field dominated by polar vortices—as previously reported by Yoden and Yamada. In contrast, the case with a finite deformation radius (i.e., the full spherical shallow‐water system) spontaneously evolves toward a banded configuration, the number of bands increasing with the rotation rate. A direct application of this shallow‐water model to the Jovian atmosphere is discussed. Using standard values for the planetary radius and rotation, we show how the initially turbulent flow self‐organizes into a potential vorticity field containing zonal structures, where regions of steep potential vorticity gradients (jets) separate relatively homogenized bands. Moreover, Jovian parameter values in our simulations lead to a strong vorticity asymmetry, favoring anticyclonic vortices—in further agreement with observations.Geophysics, Atmospheric sciences, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesTime-Dependent Fully Nonlinear Geostrophic Adjustment
http://academiccommons.columbia.edu/catalog/ac:167071
Kuo, Allen C.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22170Fri, 08 Nov 2013 00:00:00 +0000Shock-capturing numerical methods are employed to integrate the fully nonlinear, rotating 1D shallow-water equations starting from steplike nongeostrophic initial conditions (a Rossby adjustment problem). Such numerical methods allow one to observe the formation of multiple bores during the transient adjustment process as well as their decay due to rotation. It is demonstrated that increasing the rotation and/or the nonlinearity increases the rate of decay. Additionally, the time required for adjustment to be completed and its dependence on nonlinearity is examined; this time is found to be highly measure dependent. Lastly, the final adjusted state of the system is observed through long time integrations. Although the bores that form provide a mechanism for dissipation, their decay results in a final state in very good agreement with the one computed by well-known (dissipationless) conservation methods.Physics, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesThe Contour-Advective Semi-Lagrangian Algorithm for the Shallow Water Equations
http://academiccommons.columbia.edu/catalog/ac:167065
Dritschel, David G.; Polvani, Lorenzo M.; Mohebalhojeh, Ali R.http://hdl.handle.net/10022/AC:P:22168Fri, 08 Nov 2013 00:00:00 +0000A new method for integrating shallow water equations, the contour-advective semi-Lagrangian (CASL) algorithm, is presented. This is the first implementation of a contour method to a system of equations for which exact potential vorticity invertibility does not exist. The new CASL method fuses the recent contour-advection technique with the traditional pseudospectral (PS) method. The potential vorticity field, which typically develops steep gradients and evolves into thin filaments, is discretized by level sets separated by contours that are advected in a fully Lagrangian way. The height and divergence fields, which are intrinsically broader in scale, are treated in an Eulerian way: they are discretized on an fixed grid and time stepped with a PS scheme. In fact, the CASL method is similar to the widely used semi-Lagrangian (SL) method in that material conservation of potential vorticity along particle trajectories is used to determine the potential vorticity at each time step from the previous one. The crucial difference is that, whereas in the CASL method the potential vorticity is merely advected, in the SL method the potential vorticity needs to be interpolated at each time step. This interpolation results in numerical diffusion in the SL method. By directly comparing the CASL, SL, and PS methods, it is demonstrated that the implicit diffusion associated with potential vorticity interpolation in the SL method and the explicit diffusion required for numerical stability in the PS method seriously degrade the solution accuracy compared with the CASL method. Moreover, it is shown that the CASL method is much more efficient than the SL and PS methods since, for a given solution accuracy, a much coarser grid can be used and hence much faster computations can be performed.Atmospheric sciences, Applied mathematics, Geophysicslmp3Applied Physics and Applied MathematicsArticlesThe Three-Dimensional Structure of Breaking Rossby Waves in the Polar Wintertime Stratosphere
http://academiccommons.columbia.edu/catalog/ac:167053
Polvani, Lorenzo M.; Saravanan, R.http://hdl.handle.net/10022/AC:P:22164Fri, 08 Nov 2013 00:00:00 +0000The three-dimensional nature of breaking Rossby waves in the polar wintertime stratosphere is studied using an idealized global primitive equation model. The model is initialized with a well-formed polar vortex, characterized by a latitudinal band of steep potential vorticity (PV) gradients. Planetary-scale Rossby waves are generated by varying the topography of the bottom boundary, corresponding to undulations of the tropopause. Such topographically forced Rossby waves then propagate up the edge of the vortex, and their amplification with height leads to irreversible wave breaking. These numerical experiments highlight several nonlinear aspects of stratospheric dynamics that are beyond the reach of both isentropic two-dimensional models and fully realistic GCM simulations. They also show that the polar vortex is contorted by the breaking Rossby waves in a surprisingly wide range of shapes. With zonal wavenumber-1 forcing, wave breaking usually initiates as a deep helical tongue of PV that is extruded from the polar vortex. This tongue is often observed to roll up into deep isolated columns, which, in turn, may be stretched and tilted by horizontal and vertical shears. The wave amplitude directly controls the depth of the wave breaking region and the amount of vortex erosion. At large forcing amplitudes, the wave breaking in the middle/lower portions of the vortex destroys the PV gradients essential for vertical propagation, thus shielding the top of the vortex from further wave breaking. The initial vertical structure of the polar vortex is shown to play an important role in determining the characteristics of the wave breaking. Perhaps surprisingly, initially steeper PV gradients allow for stronger vertical wave propagation and thus lead to stronger erosion. Vertical wind shear has the notable effect of tilting and stretching PV structures, and thus dramatically accelerating the downscale stirring. An initial decrease in vortex area with increasing height (i.e., a conical shape) leads to focusing of wave activity, which amplifies the wave breaking. This effect provides a geometric interpretation of the “preconditioning” that often precedes a stratospheric sudden warming event. The implications for stratospheric dynamics of these and other three-dimensional vortex properties are discussed.Atmospheric sciences, Aeronomy, Geophysicslmp3Applied Physics and Applied MathematicsArticlesThe Coherent Structures of Shallowwater Turbulence: Deformationradius Effects, Cyclone/Anticyclone Asymmetry and Gravitywave Generation
http://academiccommons.columbia.edu/catalog/ac:167083
Polvani, Lorenzo M.; McWilliams, J. C.; Spall, M. A.; Ford, R.Fri, 08 Nov 2013 00:00:00 +0000Over a large range of Rossby and Froude numbers, we investigate the dynamics of initially balanced decaying turbulence in a shallow rotating fluid layer. As in the case of incompressible two‐dimensional decaying turbulence, coherent vortex structures spontaneously emerge from the initially random flow. However, owing to the presence of a free surface, a wealth of new phenomena appear in the shallow‐water system. The upscale energy cascade, common to strongly rotating flows, is arrested by the presence of a finite Rossby deformation radius. Moreover, in contrast to near‐geostrophic dynamics, a strong asymmetry is observed to develop as the Froude number is increased, leading to a clear dominance of anticyclonic vortices over cyclonic ones, even though no β effect is present in the system. Finally, we observe gravity waves to be generated around the vortex structures, and, in the strongest cases, they appear in the form of shocks. We briefly discuss the relevance of this study to the vortices observed in Jupiter’s atmosphere.Atmospheric sciences, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesReply
http://academiccommons.columbia.edu/catalog/ac:167077
Polvani, Lorenzo M.; Waugh, D. W.; Plumb, R. Alanhttp://hdl.handle.net/10022/AC:P:22172Fri, 08 Nov 2013 00:00:00 +0000Reply to Juckes about the underlying dynamics of an edge in the subtropical statosphere that mirror the "sharp, poleward 'edge' to the stratospheric surf zones as a direct consequence of the termination at those edges of the breaking of quasi-stationary Rossby waves".Atmospheric sciences, Aeronomy, Geophysicslmp3Applied Physics and Applied MathematicsArticlesOn the Subtropical Edge of the Stratospheric Surf Zone
http://academiccommons.columbia.edu/catalog/ac:167080
Polvani, Lorenzo M.; Waugh, D. W.; Plumb, R. Alanhttp://hdl.handle.net/10022/AC:P:22173Fri, 08 Nov 2013 00:00:00 +0000The formation of a subtropical “transport barrier” in the wintertime stratosphere is investigated in the context of a high-resolution shallow-water model in which Rossby waves are topographically forced on a zonally symmetric basic state. Two sets of experiments are performed: in the first “adiabatic” set, no dissipation or forcing of the mean state is imposed; in the second set, the layer thickness is relaxed to an equilibrium state taken to be representative of middle stratospheric radiative equilibrium temperatures. It is found that in the adiabatic case only a very weak subtropical barrier forms for forcing amplitudes that generate realistically steep potential vorticity gradients at the edge of the polar vortex; the vigorous wave breaking in the surf zone generates secondary waves that spread and, in turn, break well into the summer hemisphere. In contrast, the inclusion of relaxation to a realistic thermal equilibrium leads to the formation of a subtropical region of steep PV gradients. The strong subtropical shear induced by die diabatic relaxation is shown to be an important factor for the formation of the subtropical edge of the surf zone. Furthermore, the authors demonstrate that a simple one-layer shallow-water model can capture the full process of the formation of a surf zone with both polar and tropical edges starting from conditions typical of the early fall–that is, with a flow in which the polar vortex is not initially present. Finally, the authors quantify the mixing of polar and subtropical air into the midlatitude surf zone with the help of the contour advection technique. Although the quantitative estimates depend sensitively on how the edges of the surf zone are defined, our results indicate that more tropical than polar air is entrained into the surf zone.Atmospheric sciences, Aeronomy, Geophysicslmp3Applied Physics and Applied MathematicsArticlesThe Effect of a Hadley Circulation on the Propagation and Reflection of Planetary Waves in a Simple One-Layer Model
http://academiccommons.columbia.edu/catalog/ac:167050
Esler, J. Gavin; Polvani, Lorenzo M.; Plumb, R. Alanhttp://hdl.handle.net/10022/AC:P:22163Fri, 08 Nov 2013 00:00:00 +0000The effect of a simple representation of the Hadley circulation on the propagation and nonlinear reflection of planetary-scale Rossby waves in the winter hemisphere is investigated numerically in a single-layer shallow-water model. In the first instance, waves are forced by a zonal wavenumber three topography centered in the extratropics. In the linear limit the location of the low-latitude critical line at which the waves are absorbed is displaced poleward by the Hadley circulation. At finite forcing amplitude the critical layer regions where the waves break are found to be displaced poleward by a similar distance. The Hadley circulation is also found to inhibit the onset of nonlinear reflection by increasing the dissipation of wave activity in the critical layer. Second, for waves generated by an isolated mountain, the presence of the Hadley circulation further inhibits nonlinear reflection by generating a strong westerly flux of wave activity within the critical layer. This westerly flux is shown to be largely advective and is explained by the poleward displacement of the critical line into the region of westerly flow. A simple expression is derived for the minimum zonal wind strength allowing propagation in the case of a quasigeostrophic β-plane flow when the mean meridional wind ̅υ greater than 0.Geophysics, Atmospheric sciences, Physicslmp3Applied Physics and Applied MathematicsArticlesTime Variability and Simmons–Wallace–Branstator Instability in a Simple Nonlinear One-Layer Model
http://academiccommons.columbia.edu/catalog/ac:167062
Polvani, Lorenzo M.; Esler, J. Gavin; Plumb, R. Alanhttp://hdl.handle.net/10022/AC:P:22167Fri, 08 Nov 2013 00:00:00 +0000Using a global, one-layer shallow water model, the response of a westerly flow to a localized mountain is investigated. A steady, linear response at small mountain heights successively gives way first to a steady flow in which nonlinearities are important and then to unsteady, but periodic, flow at larger mountain heights. At first the unsteady behavior consists of a low-frequency oscillation of the entire Northern Hemisphere zonal flow. As the mountain height is increased further, however, the oscillatory behavior becomes localized in the diffluent jet exit region downstream of the mountain. The oscillation then takes the form of a relatively rapid vortex shedding event, followed by a gradual readjustment of the split jet structure in the diffluent region. Although relatively simple, the model exhibits a surprisingly high sensitivity to slight parameter changes. A linear stability analysis of the time-averaged flow is able to capture the transition from steady to time-dependent behavior, but fails to capture the transition between the two distinct regimes of time-dependent response. Moreover, the most unstable modes of the time-averaged flow are found to be stationary and fail to capture the salient features of the EOFs of the full time-dependent flow. These results therefore suggest that, even in the simplest cases, such as the one studied here, a linear analysis of the time-averaged flow can be highly inadequate in describing the full nonlinear behavior.Atmospheric sciences, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesOn the Mix-Down Times of Dynamically Active Potential Vorticity Filaments
http://academiccommons.columbia.edu/catalog/ac:167059
Esler, J. Gavin; Polvani, Lorenzo M.; Plumb, R. Alanhttp://hdl.handle.net/10022/AC:P:22166Fri, 08 Nov 2013 00:00:00 +0000A simple model is used to study the evolution of potential vorticity filaments, viewed in cross-section, subject to steady shear and deformation flows representative of the large-scale atmospheric circulation. It is found that the balanced,ageostrophic circulation induced by the anomalous potential vorticity can cause the evolution of a dynamically active filament to differ substantially from that of a dynamically passive filament in a similar background flow. It is suggested that estimates of the mix-down time of material contained in atmospheric filaments need to be corrected to allow for this effect.Atmospheric sciences, Geophysics, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesClimatology of Intrusions into the Tropical Upper Troposphere
http://academiccommons.columbia.edu/catalog/ac:167044
Waugh, Darryn W.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22161Thu, 07 Nov 2013 00:00:00 +0000Regions of upper tropospheric equatorial westerly winds, observed over the Pacific and Atlantic Oceans during northern fall to spring, are important for extratropical-tropical interactions. This paper focuses on one feature of these “westerly ducts” that has received relatively little attention to date: the occurrence of Rossby wave breaking events that transport tongues of extratropical air deep into the tropics, mix tropical and subtropical air, and can affect deep convection. A climatology of these “intrusion” events formed from 20 years of meteorological analyses shows a strong dependence on the basic-state flow. Notably, intrusion events are found to occur almost exclusively within westerly ducts, with more events in the presence of stronger equatorial westerlies. It is also found that there is strong interannual variability in the frequency of Pacific events, with fewer events during the warm phases of ENSO (consistent with the changes in the basic flow). Since these intrusion events laterally mix trace constituents and have been linked to tropical convection, their spatial and temporal variability may cause related variability in the distribution of trace constituents and tropical convection.Atmospheric sciences, Aeronomy, Meteorologylmp3Applied Physics and Applied MathematicsArticlesInternal Variability of the Winter Stratosphere. Part II: Time-Dependent Forcing
http://academiccommons.columbia.edu/catalog/ac:166906
Scott, R. K.; Polvani, Lorenzo M.; Waugh, D. W.http://hdl.handle.net/10022/AC:P:22107Mon, 04 Nov 2013 00:00:00 +0000This paper considers the effect of time-dependent lower boundary wave forcing on the internal variability found to appear spontaneously in a stratosphere-only model when the forcing is perfectly steady. While the time-dependent forcing is found to modulate the internal variability, leading in some cases to frequency locking of the upper-stratospheric response to the forcing, the temporal and spatial structure of the variability remains similar to the case when the forcing is time independent. Experiments with a time-periodic modulation of the forcing amplitude indicate that the wave flux through the lower boundary is only partially related to the instantaneous forcing, but is more significantly influenced by the condition of the polar vortex itself. In cases of purely random wave forcing with zero time mean, the stratospheric response is similar to that obtained with steady forcing of magnitude equal to the root-mean-square of the time-varying forcing.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesForced-Dissipative Shallow-Water Turbulence on the Sphere and the Atmospheric Circulation of the Giant Planets
http://academiccommons.columbia.edu/catalog/ac:166921
Scott, R. K.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22112Mon, 04 Nov 2013 00:00:00 +0000Although possibly the simplest model for the atmospheres of the giant planets, the turbulent forced-dissipative shallow-water system in spherical geometry has not, to date, been investigated; the present study aims to fill this gap. Unlike the freely decaying shallow-water system described by Cho and Polvani, equilibrium states in the forced-dissipative system are highly dependent on details of the forcing and the dissipation. For instance, it is found that for a given equilibrated energy level, the steadiness of zonal jets depends crucially on the balance between forcing and dissipation. With long (up to 100 000 days) high-resolution (T170) calculations, the dependence of the equilibrium states on Rossby number Ro and Rossby deformation radius LD is explored, for the case when the dissipation takes the form of hypodiffusion (acting predominantly at large scales) and the random forcing at small scales is δ correlated in time. When LD is large compared to the planetary radius, zonal jets are verified to scale closely with the Rhines scale over a wide range of Ro; furthermore, the jets at the equator are found to be both prograde and retrograde with approximately equal likelihood. As LD is decreased, the equatorial jets become increasingly and consistently retrograde, in agreement with the freely decaying turbulence results. Also, the regime recently discussed by Theiss, where zonal jets are confined to low latitudes, is illustrated to emerge robustly in the limit of small LD. Finally, specific calculations with parameter values typical of the giant planets are presented, confirming many of the earlier results obtained in the freely decaying case.Atmospheric sciences, Applied mathematics, Planetologylmp3Applied Physics and Applied MathematicsArticlesThe Antarctic Stratospheric Sudden Warming of 2002: A Self-Tuned Resonance?
http://academiccommons.columbia.edu/catalog/ac:166947
Esler, J. G.; Polvani, Lorenzo M.; Scott, R. K.http://hdl.handle.net/10022/AC:P:22124Mon, 04 Nov 2013 00:00:00 +0000The extraordinary Antarctic stratospheric warming event of 2002 was characterized by a remarkable vertical structure, with the vortex observed to divide at upper levels in the stratosphere but not at lower levels: such ‘partially’ split vortex events are relatively rare. A simple, yet fully three-dimensional, model is constructed to investigate the dynamics of this unique event. Planetary waves are excited on the model vortex edge by a lower boundary forcing characterized by two parameters: an amplitude hF and a frequency ωF, measured relative to a stationary frame. For realistic forcing amplitudes, a partial vortex split resembling that observed during the 2002 event is found only within a specific, narrow band of forcing frequencies. Exploiting the relative simplicity of our model, these frequencies are shown to be those causing a ‘self-tuning’ resonant excitation of the gravest linear mode, during which nonlinear feedback causes an initially off-resonant forcing to approach resonance.Atmospheric sciences, Aeronomy, Climate changelmp3Applied Physics and Applied MathematicsArticlesThe Coupled Stratosphere–Troposphere Response to Impulsive Forcing from the Troposphere
http://academiccommons.columbia.edu/catalog/ac:166959
Reichler, Thomas; Kushner, Paul J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22128Mon, 04 Nov 2013 00:00:00 +0000A simple atmospheric general circulation model (GCM) is used to investigate the transient response of the stratosphere–troposphere system to externally imposed pulses of lower-tropospheric planetary wave activity. The atmospheric GCM is a dry, hydrostatic, global primitive-equations model, whose circulation includes an active polar vortex and a tropospheric jet maintained by baroclinic eddies. Planetary wave activity pulses are generated by a perturbation of the solid lower boundary that grow and decay over a period of 10 days. The planetary wave pulses propagate upward and break in the stratosphere. Subsequently, a zonal-mean circulation anomaly propagates downward, often into the troposphere, at lags of 30–100 days. The evolution of the response is found to be dependent on the state of the stratosphere–troposphere system at the time the pulse is generated. In particular, on the basis of a large ensemble of these simulations, it is found that the length of time the signal takes to propagate downward from the stratosphere is controlled by initial anomalies in the zonal-mean circulation and in the zonal-mean wave drag. Criteria based on these anomaly patterns can be used, therefore, to predict the long-term surface response of the stratosphere–troposphere system to a planetary wave pulse up to 90 days after the pulse is generated. In an independent test, it is verified that the initial states that most strongly satisfy these criteria respond in the expected way to the lower-tropospheric wave activity pulse.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesTesting the Annular Mode Autocorrelation Time Scale in Simple Atmospheric General Circulation Models
http://academiccommons.columbia.edu/catalog/ac:166909
Gerber, Edwin P.; Voronin, Sergey; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22108Mon, 04 Nov 2013 00:00:00 +0000A new diagnostic for measuring the ability of atmospheric models to reproduce realistic low-frequency variability is introduced in the context of Held and Suarez’s 1994 proposal for comparing the dynamics of different general circulation models. A simple procedure to compute τ, the e-folding time scale of the annular mode autocorrelation function, is presented. This quantity concisely quantifies the strength of low-frequency variability in a model and is easy to compute in practice. The sensitivity of τ to model numerics is then studied for two dry primitive equation models driven with the Held–Suarez forcings: one pseudospectral and the other finite volume. For both models, τ is found to be unrealistically large when the horizontal resolutions are low, such as those that are often used in studies in which long integrations are needed to analyze model variability on low frequencies. More surprising is that it is found that, for the pseudospectral model, τ is particularly sensitive to vertical resolution, especially with a triangular truncation at wavenumber 42 (a very common resolution choice). At sufficiently high resolution, the annular mode autocorrelation time scale τ in both models appears to converge around values of 20–25 days, suggesting the existence of an intrinsic time scale at which the extratropical jet vacillates in the Held and Suarez system. The importance of τ for computing the correct response of a model to climate change is explicitly demonstrated by perturbing the pseudospectral model with simple torques. The amplitude of the model’s response to external forcing increases as τ increases, as suggested by the fluctuation–dissipation theorem.Atmospheric sciences, Meteorology, Applied mathematicsepg2108, sv2122, lmp3Applied Physics and Applied MathematicsArticlesEnhancement of Rossby Wave Breaking by Steep Potential Vorticity Gradients in the Winter Stratosphere
http://academiccommons.columbia.edu/catalog/ac:166977
Scott, R. K.; Dritschel, D. G.; Polvani, Lorenzo M.; Waugh, D. W.http://hdl.handle.net/10022/AC:P:22134Mon, 04 Nov 2013 00:00:00 +0000This work investigates the extent to which potential vorticity gradients affect the vertical propagation of planetary-scale Rossby waves on the edge of a stratospheric polar vortex and their eventual nonlinear saturation and breaking. Using two different numerical modeling approaches, it is shown that wave propagation and wave breaking are significantly reduced when the potential vorticity gradients at the vortex edge are anomalously weak. The efficiency of the first model, based on high-resolution contour dynamics, permits a full exploration of the parameter space of wave forcing amplitude and edge steepness. A more realistic primitive equation model in spherical geometry both confirms the contour dynamics results and highlights some direct implications for stratospheric modeling in more comprehensive models. The results suggest that stratospheric models using horizontal resolutions of spectral T42 or less may significantly underestimate the vertical propagation and breaking of planetary waves, and may consequently misrepresent such important stratospheric processes as the mean meridional circulation, sudden warmings, and the mixing of chemically distinct polar and midlatitude air.Atmospheric sciences, Aeronomy, Atmospheric chemistrylmp3Applied Physics and Applied MathematicsArticlesStratospheric Control of Upward Wave Flux near the Tropopause
http://academiccommons.columbia.edu/catalog/ac:166965
Scott, R. K.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22130Mon, 04 Nov 2013 00:00:00 +0000Using an idealized, global primitive equation model of the stratosphere-troposphere system in which all tropospheric variability is surpressed, we demonstrate the existence of internal modes of stratospheric variability. The variability in our model is similar to that observed in the winter stratosphere, consisting of sudden-warming like, wave-driven decelerations of the polar vortex followed by a more gradual re-establishment of the vortex by the radiative forcing. Using a common index of the strength of the stratospheric vortex, we find patterns of downward propagation resembling those found in recent observations. In addition, our model exhibits considerable variability in the upward flux of wave activity into the stratosphere; this variability strongly anti-correlates with the index of the mid-upper stratospheric vortex, again in agreement with recent observations.Atmospheric sciences, Applied mathematics, Aeronomylmp3Applied Physics and Applied MathematicsArticlesInternal Variability of the Winter Stratosphere. Part I: Time-Independent Forcing
http://academiccommons.columbia.edu/catalog/ac:166950
Scott, R. K.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22125Mon, 04 Nov 2013 00:00:00 +0000This paper examines the nature and robustness of internal stratospheric variability, namely the variability resulting from the internal dynamics of the stratosphere itself, as opposed to that forced by external sources such as the natural variability of the free troposphere. Internal stratospheric variability arises from the competing actions of radiative forcing, which under perpetual winter conditions strengthens the polar vortex, and planetary wave breaking, which weakens it. The results from a stratosphere-only model demonstrate that strong internal stratospheric variability, consisting of repeated sudden warming-type events, exists over a wide range of realistic radiative and wave forcing conditions, and is largely independent of other physical and numerical parameters. In particular, the coherent form of the variability persists as the number of degrees of freedom is increased, and is therefore not an artifact of severe model truncation. Various diagnostics, including three-dimensional representations of the potential vorticity, illustrate that the variability is determined by the vertical structure of the vortex and the extent to which upward wave propagation is favored or inhibited. In this paper, the variability arising from purely internal stratosphere dynamics is isolated by specifying thermal and wave forcings that are completely time independent. In a second paper, the authors investigate the relative importance of internal and external variability by considering time-dependent wave forcing as a simple representation of tropospheric variability.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesKelvin–Helmholtz Instability of Potential Vorticity Layers: A Route to Mixing
http://academiccommons.columbia.edu/catalog/ac:166980
Esler, J. G.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22135Mon, 04 Nov 2013 00:00:00 +0000The linear and nonlinear dynamics of layers of anomalously high potential vorticity (PV) are studied in detail. It is well known that PV layers are subject to slow, balanced, mixed barotropic–baroclinic instabilities. In this paper, it is shown that, in addition, PV layers are subject to a Kelvin–Helmholtz instability, operating on much smaller spatial and faster temporal scales. For simplicity, spatially infinite layers of uniform anomalous PV are considered. Such layers are characterized by two key parameters: the ratio Δq of their anomalous PV to the background PV, and the angle α between the layer and the direction of the ambient stratification gradient (in suitably scaled coordinates). It is found that Kelvin–Helmholtz appears, for certain values of α, whenever Δq greater than 8. Of notable interest is the case of an initially vertical PV layer embedded in a weak ambient shear flow: for sufficiently large Δq, once the PV layer is tilted past a critical angle, Kelvin–Helmholtz instability becomes possible. It is argued that the breakdown of PV layers due to a Kelvin–Helmholtz instability induced by ambient shear might be an important systematic mechanism leading to irreversible mixing during stratosphere–troposphere exchange events. This is discussed in the context of an example of Kelvin–Helmholtz instability observed near a tropopause fold.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesA New Look at Stratospheric Sudden Warmings. Part III: Polar Vortex Evolution and Vertical Structure
http://academiccommons.columbia.edu/catalog/ac:166884
Matthewman, N. J.; Esler, J. G.; Charlton-Perez, A. J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22100Fri, 01 Nov 2013 00:00:00 +0000The evolution of the Arctic polar vortex during observed major midwinter stratospheric sudden warmings (SSWs) is investigated for the period 1957–2002, using 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) Ertel’s potential vorticity (PV) and temperature fields. Time-lag composites of vertically weighted PV, calculated relative to the SSW onset time, are derived for both vortex-displacement SSWs and vortex-splitting SSWs, by averaging over the 15 recorded displacement and 13 splitting events. The evolving vertical structure of the polar vortex during a typical SSW of each type is clearly illustrated by plotting an isosurface of the composite PV field, and is shown to be very close to that observed during representative individual events. Results are verified by comparison with an elliptical diagnostic vortex moment technique. For both types of SSW, little variation is found between individual events in the orientation of the developing vortex relative to the underlying topography; that is, the location of the vortex during SSWs of each type is largely fixed in relation to the earth’s surface. During each type of SSW, the vortex is found to have a distinctive vertical structure. Vortex-splitting events are typically barotropic, with the vortex split occurring near simultaneously over a large altitude range (20–40 km). In the majority of cases, of the two daughter vortices formed, it is the “Siberian” vortex that dominates over its “Canadian” counterpart. In contrast, displacement events are characterized by a very clear baroclinic structure; the vortex tilts significantly westward with height, so that the top and bottom of the vortex are separated by nearly 180° longitude before the upper vortex is sheared away and destroyed.Atmospheric sciences, Meteorology, Aeronomylmp3Applied Physics and Applied MathematicsArticlesA New Look at Stratospheric Sudden Warmings. Part III: Polar Vortex Evolution and Vertical Structure
http://academiccommons.columbia.edu/catalog/ac:166887
Matthewman, N. J.; Esler, J. G.; Charlton-Perez, A. J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22100Fri, 01 Nov 2013 00:00:00 +0000The evolution of the Arctic polar vortex during observed major midwinter stratospheric sudden warmings (SSWs) is investigated for the period 1957–2002, using 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) Ertel’s potential vorticity (PV) and temperature fields. Time-lag composites of vertically weighted PV, calculated relative to the SSW onset time, are derived for both vortex-displacement SSWs and vortex-splitting SSWs, by averaging over the 15 recorded displacement and 13 splitting events. The evolving vertical structure of the polar vortex during a typical SSW of each type is clearly illustrated by plotting an isosurface of the composite PV field, and is shown to be very close to that observed during representative individual events. Results are verified by comparison with an elliptical diagnostic vortex moment technique. For both types of SSW, little variation is found between individual events in the orientation of the developing vortex relative to the underlying topography; that is, the location of the vortex during SSWs of each type is largely fixed in relation to the earth’s surface. During each type of SSW, the vortex is found to have a distinctive vertical structure. Vortex-splitting events are typically barotropic, with the vortex split occurring near simultaneously over a large altitude range (20–40 km). In the majority of cases, of the two daughter vortices formed, it is the “Siberian” vortex that dominates over its “Canadian” counterpart. In contrast, displacement events are characterized by a very clear baroclinic structure; the vortex tilts significantly westward with height, so that the top and bottom of the vortex are separated by nearly 180° longitude before the upper vortex is sheared away and destroyed.Atmospheric sciences, Meteorology, Aeronomylmp3Applied Physics and Applied MathematicsArticlesEquatorial Superrotation in Shallow Atmospheres
http://academiccommons.columbia.edu/catalog/ac:166899
Scott, R. K.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22104Fri, 01 Nov 2013 00:00:00 +0000Simple, shallow-water models have been successful in reproducing two key observables in the atmospheres of the giant planets: the formation of robust, and fully turbulent, latitudinal jets and the decrease of the zonal wind amplitude with latitude. However, they have to date consistently failed in reproducing the strong prograde (superrotating) equatorial winds that are often observed on such planets. In this paper we show that shallow water models not only can give rise to superrotating winds, but can do so very robustly, provided that the physical process of large-scale energy dissipation by radiative relaxation is taken into account. When energy is removed by linear friction, equatorial superrotation does not develop; when energy is removed by radiative relaxation, superrotation develops at apparently any deformation radius.Atmospheric sciences, Meteorology, Applied mathematicslmp3Applied Physics and Applied MathematicsArticlesUsing satellites to investigate the sensitivity of longwave downward radiation to water vapor at high elevations
http://academiccommons.columbia.edu/catalog/ac:158973
Naud, Catherine M.; Miller, James R.; Landry, Chrishttp://hdl.handle.net/10022/AC:P:19709Tue, 09 Apr 2013 00:00:00 +0000Many studies suggest that high-elevation regions may be among the most sensitive to future climate change. However, in situ observations in these often remote locations are too sparse to determine the feedbacks responsible for enhanced warming rates. One of these feedbacks is associated with the sensitivity of longwave downward radiation (LDR) to changes in water vapor, with the sensitivity being particularly large in many high-elevation regions where the average water vapor is often low. We show that satellite retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS) and Clouds and the Earth’s Radiant Energy System (CERES) can be used to expand the current ground-based observational database and that the monthly averaged clear-sky satellite estimates of humidity and LDR are in good agreement with the well-instrumented Center for Snow and Avalanche Studies ground-based site in the southwestern Colorado Rocky Mountains. The relationship between MODIS-retrieved precipitable water vapor and surface specific humidity across the contiguous United States was found to be similar to that previously found for the Alps. More important, we show that satellites capture the nonlinear relationship between LDR and water vapor and confirm that LDR is especially sensitive to changes in water vapor at high elevations in several midlatitude mountain ranges. Because the global population depends on adequate fresh water, much of which has its source in high mountains, it is critically important to understand how climate will change there. We demonstrate that satellites can be used to investigate these feedbacks in high-elevation regions where the coverage of surface-based observations is insufficient to do so.Climate change, Atmospheric sciences, Meteorologycn2140Applied Physics and Applied MathematicsArticlesDiffraction on the Two-Dimensional Square Lattice
http://academiccommons.columbia.edu/catalog/ac:134447
Bhat, H. S.; Osting, Braxtonhttp://hdl.handle.net/10022/AC:P:10563Tue, 21 Jun 2011 00:00:00 +0000We solve the thin-slit diffraction problem for two-dimensional lattice waves. More precisely, for the discrete Helmholtz equation on the semi-infinite square lattice with data prescribed on the left boundary (the aperture), we use lattice Green's functions and a discrete Sommerfeld outgoing radiation condition to derive the exact solution everywhere in the lattice. The solution is a discrete convolution that can be evaluated in closed form for the wave number $k=2$. For other wave numbers, we give a recursive algorithm for computing the convolution kernel.Applied mathematicsbro2103Applied Physics and Applied MathematicsArticles