Academic Commons Search Results
http://academiccommons.columbia.edu/catalog.rss?f%5Bsubject_facet%5D%5B%5D=Applied+mathematics&q=&rows=500&sort=record_creation_date+desc
Academic Commons Search Resultsen-usEnlargement of Filtration and the Strict Local Martingale Property in Stochastic Differential Equations
http://academiccommons.columbia.edu/catalog/ac:201869
Dandapani, Aditihttp://dx.doi.org/10.7916/D8XW4JZ2Tue, 02 Aug 2016 12:25:23 +0000In this thesis, we study the strict local martingale property of solutions of various types of stochastic differential equations and the effect of an initial expansion of the filtration on this property. For the models we consider, we either use existing criteria or, in the case where the stochastic differential equation has jumps, develop new criteria that can can detect the presence of the strict local martingale property. We develop deterministic sufficient conditions on the drift and diffusion coefficient of the stochastic process such that an enlargement by initial expansion of the filtration can produce a strict local martingale from a true martingale. We also develop a way of characterizing the martingale property in stochastic volatility models where the local martingale has a general diffusion coefficient.Mathematics, Applied mathematicsad2259Applied Physics and Applied Mathematics, StatisticsDissertationsBifurcation perspective on topologically protected and non-protected states in continuous systems
http://academiccommons.columbia.edu/catalog/ac:197647
Lee-Thorp, James Patrickhttp://dx.doi.org/10.7916/D8J38SJRThu, 14 Apr 2016 12:21:18 +0000We study Schrödinger operators perturbed by non-compact (spatially extended) defects. We consider two models: a one-dimensional (1D) dimer structure with a global phase shift, and a two-dimensional (2D) honeycomb structure with a line-defect or "edge''. In both the 1D and 2D settings, the non-compact defects are modeled by adiabatic, domain wall modulations of the respective dimer and honeycomb structures. Our main results relate to the rigorous construction of states via bifurcations from continuous spectra. These bifurcations are controlled by asymptotic effective (homogenized) equations that underlie the protected or non-protected character of the states.
In 1D, the states we construct are localized solutions. In 2D, they are "edge states'' - time-harmonic solutions which are propagating (plane-wave-like) parallel to a line-defect or "edge'' and are localized transverse to it. The states are described as protected if they persist in the presence of spatially localized (even strong) deformations of the global phase defect (in 1D) or edge (in 2D). The protected states bifurcate from "Dirac points'' (linear/conical spectral band-crossings) in the continuous spectra and are seeded by an effective Dirac equation. The (more conventional) non-protected states bifurcate from spectral band edges are seeded by an effective Schrödinger equation.
Our 2D model captures many aspects of the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and "artificial graphene''. The protected states we construct in our 1D dimer model can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. We present a detailed computational study of an experimentally realizable photonic waveguide array structure.Applied mathematics, Mathematics, Physicsjpl2154Applied Physics and Applied MathematicsDissertationsOptimization methods for power grid reliability
http://academiccommons.columbia.edu/catalog/ac:194133
Harnett, Sean R.http://dx.doi.org/10.7916/D8MS3SM8Thu, 04 Feb 2016 18:18:39 +0000This dissertation focuses on two specific problems related to the reliability of the modern power grid. The first part investigates the economic dispatch problem with uncertain power sources. The classic economic dispatch problem seeks generator power output levels that meet demand most efficiently; we add risk-awareness to this by explicitly modeling the uncertainty of intermittent power sources using chance-constrained optimization and incorporating the chance constraints into the standard optimal power flow framework. The result is a dispatch of power which is substantially more robust to random fluctuations with only a small increase in economic cost. Furthermore, it uses an algorithm which is only moderately slower than the conventional practice.
The second part investigates “the power grid attack problem”: aiming to maximize disruption to the grid, how should an attacker distribute a budget of “damage” across the power lines? We formulate it as a continuous problem, which bypasses the combinatorial explosion of a discrete formulation and allows for interesting attacks containing lines that are only partially damaged rather than completely removed. The result of our solution to the attack problem can provide helpful information to grid planners seeking to improve the resilience of the power grid to outages and disturbances. Both parts of this dissertation include extensive experimental results on a number of cases, including many realistic large-scale instances.Applied mathematics, Operations researchsrh2144Applied Physics and Applied MathematicsDissertationsDistributed and Large-Scale Optimization
http://academiccommons.columbia.edu/catalog/ac:193921
Ali Younis Kalbat, Abdulrahman Younishttp://dx.doi.org/10.7916/D8D79B7VWed, 27 Jan 2016 00:00:00 +0000This dissertation is motivated by the pressing need for solving real-world large-scale optimization problems with the main objective of developing scalable algorithms that are capable of solving such problems efficiently. Large-scale optimization problems naturally appear in complex systems such as power networks and distributed control systems, which are the main systems of interest in this work. This dissertation aims to address four problems with regards to the theory and application of large-scale optimization problems, which are explained below: Chapter 2: In this chapter, a fast and parallelizable algorithm is developed for an arbitrary decomposable semidefinite program (SDP). Based on the alternating direction method of multipliers, we design a numerical algorithm that has a guaranteed convergence under very mild assumptions. We show that each iteration of this algorithm has a simple closed-form solution, consisting of matrix multiplications and eigenvalue decompositions performed by individual agents as well as information exchanges between neighboring agents. The cheap iterations of the proposed algorithm enable solving a wide spectrum of real-world large-scale conic optimization problems that could be reformulated as SDP. Chapter 3: Motivated by the application of sparse SDPs to power networks, the objective of this chapter is to design a fast and parallelizable algorithm for solving the SDP relaxation of a large-scale optimal power flow (OPF) problem. OPF is fundamental problem used for the operation and planning of power networks, which is non-convex and NP-hard in the worst case. The proposed algorithm would enable a real-time power network management and improve the system's reliability. In particular, this algorithm helps with the realization of Smart Grid by allowing to make optimal decisions very fast in response to the stochastic nature of renewable energy. The proposed algorithm is evaluated on IEEE benchmark systems. Chapter 4: The design of an optimal distributed controller using an efficient computational method is one of the most fundamental problems in the area of control systems, which remains as an open problem due to its NP-hardness in the worst case. In this chapter, we first study the infinite-horizon optimal distributed control (ODC) problem (for deterministic systems) and then generalize the results to a stochastic ODC problem (for stochastic systems). Our approach rests on formulating each of these problems as a rank-constrained optimization from which an SDP relaxation can be derived. We show that both problems admit sparse SDP relaxations with solutions of rank at most~3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the rank-3 solution may be deployed to retrieve a near-global controller. We also propose computationally cheap SDP relaxation for each problem and then develop effective heuristic methods to recover a near-optimal controller from the low-rank SDP solution. The design of several near-optimal structured controllers with global optimality degrees above 99\% will be demonstrated. Chapter 5: The frequency control problem in power networks aims to control the global frequency of the system within a tight range by adjusting the output of generators in response to the uncertain and stochastic demand. The intermittent nature of distributed power generation in smart grid makes the traditional decentralized frequency controllers less efficient and demands distributed controllers that are able to deal with the uncertainty in the system introduced by non-dispatchable supplies (such as renewable energy), fluctuating loads, and measurement noise. Motivated by this need, we study the frequency control problem using the results developed in Chapter 4. In particular, we formulate the problem and then conduct a case study on the IEEE 39-Bus New England system. The objective is to design a near-global optimal distributed frequency controller for the New England test system by optimally adjusting the mechanical power input to each generator based on the real-time measurement received from neighboring generators through a user-defined communication topology.Electrical engineering, Applied mathematics, Computer science, Semidefinite programming, Mathematical optimization, Mathematical optimization--Methodology, Computer algorithmsak3369Electrical EngineeringDissertationsSimultaneous Parameter and Input Estimation of a Respiratory Mechanics Model
http://academiccommons.columbia.edu/catalog/ac:192808
Vicario, Francesco; Albanese, Antonio; Wang, Dong; Karamolegkos, Nikolaos; Chbat, Nicolas W.http://dx.doi.org/10.7916/D88K78T3Fri, 08 Jan 2016 00:00:00 +0000Real-time noninvasive estimation of respiratory mechanics in spontaneously breathing patients is still an open problem in the field of critical care. Even assuming that the system is a simplistic first-order single-compartment model, the presence of unmeasured patient effort still makes the problem complex since both the parameters and part of the input are unknown. This paper presents an approach to overcome the underdetermined nature of the mathematical problem by infusing physiological knowledge into the estimation process and using it to construct an optimization problem subject to physiological constraints. As it relies only on measurements available on standard ventilators, namely the flow and pressure at the patient’s airway opening, the approach is noninvasive. Additionally, breath by breath, it continually provides estimates of the patient respiratory resistance and elastance as well as of the muscle effort waveform without requiring maneuvers that would interfere with the desired ventilation pattern.Mechanical engineering, Applied mathematics, Computer science, Respiration--Measurement, Respiration--Physiological aspects, Respiration--Mathematical modelsfv2157, nk2440, nc22Mechanical Engineering, Biomedical EngineeringConferencesAn All-Interaction Matrix Approach to Linear and Bilinear System Identification
http://academiccommons.columbia.edu/catalog/ac:192641
Phan, Minh Q.; Vicario, Francesco; Longman, Richard W.; Betti, Raimondohttp://dx.doi.org/10.7916/D8DV1JMFTue, 05 Jan 2016 00:00:00 +0000This paper is a brief introduction to the interaction matrices. Originally formulated as a parameter compression mechanism, the interaction matrices offer a unifying framework to treat a wide range of problems in system identification and control. We retrace the origin of the interaction matrices, and describe their applications in selected problems in system identification.Mechanical engineering, Applied mathematics, Computer science, System identification, Matrices, Markov processes--Mathematical modelsfv2157, rwl4, rb68Civil Engineering and Engineering Mechanics, Mechanical EngineeringConferencesGeneralized Framework of OKID for Linear State-Space Model Identification
http://academiccommons.columbia.edu/catalog/ac:192638
Vicario, Francesco; Phan, Minh Q.; Longman, Richard W.; Betti, Raimondohttp://dx.doi.org/10.7916/D8P84BMQTue, 05 Jan 2016 00:00:00 +0000This paper presents a generalization of observer/Kalman filter identification (OKID). OKID is a method for the simultaneous identification of a linear dynamical system and the associated Kalman filter from input-output measurements corrupted by noise. OKID was originally developed at NASA as the OKID/ERA algorithm. Recent work showed that ERA is not the only way to complete the OKID process and paved the way to the generalization of OKID as an approach to linear system identification. As opposed to other approaches, OKID is explicitly formulated via state observers providing an intuitive interpretation from a control theory perspective. The extension of the OKID framework to more complex identification problems, including nonlinear systems, is also discussed.Mechanical engineering, Applied mathematics, Computer science, State-space methods, Kalman filtering, Linear systems, Control theory--Mathematical modelsfv2157, rwl4, rb68Civil Engineering and Engineering Mechanics, Mechanical EngineeringConferencesThe Downward Influence of Stratospheric Sudden Warmings
http://academiccommons.columbia.edu/catalog/ac:190045
Hitchcock, Peter; Simpson, Isla R.http://dx.doi.org/10.7916/D8PV6JWDTue, 20 Oct 2015 14:01:57 +0000The coupling between the stratosphere and the troposphere following two major stratospheric sudden warmings is studied in the Canadian Middle Atmosphere Model using a nudging technique by which the zonal-mean evolution of the reference sudden warmings are artificially induced in an ~100-member ensemble spun off from a control simulation. Both reference warmings are taken from a freely running integration of the model. One event is a displacement, the other is a split, and both are followed by extended recoveries in the lower stratosphere. The methodology permits a statistically robust study of their influence on the troposphere below.
The nudged ensembles exhibit a tropospheric annular mode response closely analogous to that seen in observations, confirming the downward influence of sudden warmings on the troposphere in a comprehensive model. This tropospheric response coincides more closely with the lower-stratospheric annular mode anomalies than with the midstratospheric wind reversal. In addition to the expected synoptic-scale eddy feedback, the planetary-scale eddies also reinforce the tropospheric wind changes, apparently responding directly to the stratospheric anomalies.
Furthermore, despite the zonal symmetry of the stratospheric perturbation, a highly zonally asymmetric near-surface response is produced, corresponding to a strongly negative phase of the North Atlantic Oscillation with a much weaker response over the Pacific basin that matches composites of sudden warmings from the Interim ECMWF Re-Analysis (ERA-Interim). Phase 5 of the Coupled Model Intercomparison Project models exhibit a similar response, though in most models the response’s magnitude is underrepresented.Meteorology, Atmospheric sciences, Aeronomy, Applied mathematicsirs2113Lamont-Doherty Earth ObservatoryArticlesComputational Methods for Nonlinear Optimization Problems: Theory and Applications
http://academiccommons.columbia.edu/catalog/ac:189943
Madani, Ramtinhttp://dx.doi.org/10.7916/D88S4PDMThu, 15 Oct 2015 00:00:00 +0000This dissertation is motivated by the lack of efficient global optimization techniques for polynomial optimization problems. The objective is twofold. First, a new mathematical foundation for obtaining a global or near-global solution will be developed. Second, several case studies will be conducted on a variety of real-world problems. Global optimization, convex relaxation and distributed computation are at the heart of this PhD dissertation. Some of the specific problems to be addressed in this thesis on both the theory and the application of nonlinear optimization are explained below: Graph theoretic algorithms for low-rank optimization problems: There is a rapidly growing interest in the recovery of an unknown low-rank matrix from limited information and measurements. This problem occurs in many areas of engineering and applied science such as machine learning, control, and computer vision. We develop a graph-theoretic technique in Part I that is able to generate a low-rank solution for a sparse Linear Matrix Inequality (LMI), which is directly applicable to a large set of problems such as low-rank matrix completion with many unknown entries. Our approach finds a solution with a guarantee on its rank, using the recent advances in graph theory. Resource allocation for energy systems: The flows in an electrical grid are described by nonlinear AC power flow equations. Due to the nonlinear interrelation among physical parameters of the network, the feasibility region represented by power flow equations may be nonconvex and disconnected. Since 1962, the nonlinearity of the network constraints has been studied, and various heuristic and local-search algorithms have been proposed in order to perform optimization over an electrical grid [Baldick, 2006; Pandya and Joshi, 2008]. Part II is concerned with finding convex formulations of the power flow equations using semidefinite programming (SDP). The potential of SDP relaxation for problems in power systems has been manifested in [Lavaei and Low, 2012], with further studies conducted in [Lavaei, 2011; Sojoudi and Lavaei, 2012]. A variety of graph-theoretic and algebraic methods are developed in Part II in order to facilitate performing fundamental, yet challenging tasks such as optimal power flow (OPF) problem, security-constrained OPF and the classical power flow problem. Synthesis of distributed control systems: Real-world systems mostly consist of many interconnected subsystems, and designing an optimal controller for them pose several challenges to the field of control theory. The area of distributed control is created to address the challenges arising in the control of these systems. The objective is to design a constrained controller whose structure is specified by a set of permissible interactions between the local controllers with the aim of reducing the computation or communication complexity of the overall controller. It has been long known that the design of an optimal distributed (decentralized) controller is a daunting task because it amounts to an NP-hard optimization problem in general [Witsenhausen, 1968; Tsitsiklis and Athans, 1984]. Part III is devoted to study the potential of the SDP relaxation for the optimal distributed control (ODC) problem Our approach rests on formulating each of different variations of the ODC problem as rank-constrained optimization problems from which SDP relaxations can be derived. As the first contribution, we show that the ODC problem admits a sparse SDP relaxation with solutions of rank at most 3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the low-rank SDP solution may be deployed to retrieve a near-global controller. Parallel computation for sparse semidefinite programs: While small- to medium-sized semidefinite programs are efficiently solvable by second-order-based interior point methods in polynomial time up to any arbitrary precision [Vandenberghe and Boyd, 1996a], these methods are impractical for solving large-scale SDPs due to computation time and memory issues. In Part IV of this dissertation, a parallel algorithm for solving an arbitrary SDP is introduced based on the alternating direction method of multipliers. The proposed algorithm has a guaranteed convergence under very mild assumptions. Each iteration of this algorithm has a simple closed-form solution, and consists of scalar multiplication and eigenvalue decomposition over matrices whose sizes are not greater than the treewdith of the sparsity graph of the SDP problem. The cheap iterations of the proposed algorithm enable solving real-world large-scale conic optimization problems.Engineering, Applied mathematics, Computer sciencerm3122Electrical EngineeringDissertationsToward an Understanding of Vertical Momentum Transports in Cloud-System-Resolving Model Simulations of Multiscale Tropical Convection
http://academiccommons.columbia.edu/catalog/ac:189334
Shaw, Tiffany Ann; Lane, Todd P.http://dx.doi.org/10.7916/D8WD4016Mon, 12 Oct 2015 00:00:00 +0000This study examines the characteristics of convective momentum transport (CMT) and gravity wave momentum transport (GWMT) in two-dimensional cloud-system-resolving model simulations, including the relationships between the two transports. A linear group velocity criterion is shown to objectively separate CMT and GWMT. The GWMT contribution is mostly consistent with upward-propagating gravity waves and is present in the troposphere and the stratosphere. The CMT contribution forms a large part of the residual (nonupward-propagating contribution) and dominates the fluxes in the troposphere. Additional analysis of the vertical sensible heat flux supports the physical interpretation of the two contributions, further isolating the effects of unstable convection from vertically propagating gravity waves. The role of transient and nonconservative (friction and diabatic heating) processes in generating momentum flux and their dependence on changes in convective organization was assessed using a pseudomomentum budget analysis. Nonconservative effects were found to dominate the transports; the GWMT contribution involved a diabatic source region in the troposphere and a dissipative sink region in the stratosphere. The CMT contribution was consistent with transport between the boundary layer and free troposphere via tilted convection. Transient buoyancy–vorticity correlations highlighted wave sources in the region of convective outflow and the boundary layer. These sources were akin to the previously described “mechanical oscillator” mechanism. Fluxes associated with this upper-level source were most sensitive to convective organization, highlighting the mechanism by which changes in organization are communicated to GWMT. The results elucidate important interactions between CMT and GWMT, adding further weight to suggestions that the two transports should be linked in parameterizations.Atmospheric sciences, Applied mathematics, Aeronomytas2163Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesSeamless Precipitation Prediction Skill in the Tropics and Extratropics from a Global Model
http://academiccommons.columbia.edu/catalog/ac:189235
Zhu, Hongyan; Wheeler, Matthew C.; Sobel, Adam H.; Hudson, Debrahttp://dx.doi.org/10.7916/D8JM2931Thu, 08 Oct 2015 16:25:48 +0000The skill with which a coupled ocean–atmosphere model is able to predict precipitation over a range of time scales (days to months) is analyzed. For a fair comparison across the seamless range of scales, the verification is performed using data averaged over time windows equal in length to the lead time. At a lead time of 1 day, skill is greatest in the extratropics around 40°–60° latitude and lowest around 20°, and has a secondary local maximum close to the equator. The extratropical skill at this short range is highest in the winter hemisphere, presumably due to the higher predictability of winter baroclinic systems. The local equatorial maximum comes mostly from the Pacific Ocean, and thus appears to be mostly from El Niño–Southern Oscillation (ENSO). As both the lead time and averaging window are simultaneously increased, the extratropical skill drops rapidly with lead time, while the equatorial maximum remains approximately constant, causing the equatorial skill to exceed the extratropical at leads of greater than 4 days in austral summer and 1 week in boreal summer. At leads longer than 2 weeks, the extratropical skill flattens out or increases, but remains below the equatorial values. Comparisons with persistence confirm that the model beats persistence for most leads and latitudes, including for the equatorial Pacific where persistence is high. The results are consistent with the view that extratropical predictability is mostly derived from synoptic-scale atmospheric dynamics, while tropical predictability is primarily derived from the response of moist convection to slowly varying forcing such as from ENSO.Applied mathematics, Meteorology, Atmospheric sciencesahs129Applied Physics and Applied Mathematics, Earth and Environmental Sciences, Lamont-Doherty Earth ObservatoryArticlesMoisture Modes and the Eastward Propagation of the MJO
http://academiccommons.columbia.edu/catalog/ac:189151
Sobel, Adam H.; Maloney, Eric D.http://dx.doi.org/10.7916/D8H70F82Wed, 07 Oct 2015 00:00:00 +0000The authors discuss modifications to a simple linear model of intraseasonal moisture modes. Wind–evaporation feedbacks were shown in an earlier study to induce westward propagation in an eastward mean low-level flow in this model. Here additional processes, which provide effective sources of moist static energy to the disturbances and which also depend on the low-level wind, are considered. Several processes can act as positive sources in perturbation easterlies: zonal advection (if the mean zonal moisture gradient is eastward), modulation of synoptic eddy drying by the MJO-scale wind perturbations, and frictional convergence. If the sum of these is stronger than the wind–evaporation feedback—as observations suggest may be the case, though with considerable uncertainty—the model produces unstable modes that propagate weakly eastward relative to the mean flow. With a small amount of horizontal diffusion or other scale-selective damping, the growth rate is greatest at the largest horizontal scales and decreases monotonically with wavenumber.Atmospheric sciences, Applied mathematics, Meteorologyahs129Applied Physics and Applied Mathematics, Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesSimple scaling of catastrophic landslide dynamics
http://academiccommons.columbia.edu/catalog/ac:189139
Ekstrom, Goran; Stark, Colin P.http://dx.doi.org/10.7916/D8TQ610DTue, 06 Oct 2015 00:00:00 +0000Catastrophic landslides involve the acceleration and deceleration of millions of tons of rock and debris in response to the forces of gravity and dissipation. Their unpredictability and frequent location in remote areas have made observations of their dynamics rare. Through real-time detection and inverse modeling of teleseismic data, we show that landslide dynamics are primarily determined by the length scale of the source mass. When combined with geometric constraints from satellite imagery, the seismically determined landslide force histories yield estimates of landslide duration, momenta, potential energy loss, mass, and runout trajectory. Measurements of these dynamical properties for 29 teleseismogenic landslides are consistent with a simple acceleration model in which height drop and rupture depth scale with the length of the failing slope.Geology, Petrology, Applied mathematicsge21, cps13Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesMoist Static Energy Budget of the MJO during DYNAMO
http://academiccommons.columbia.edu/catalog/ac:191146
Sobel, Adam H.; Wang, Shuguang; Kim, Daehyunhttp://dx.doi.org/10.7916/D8JS9PTCMon, 28 Sep 2015 00:00:00 +0000The authors analyze the column-integrated moist static energy budget over the region of the tropical Indian Ocean covered by the sounding array during the Cooperative Indian Ocean Experiment on Intraseasonal Variability in the Year 2011 (CINDY2011)/Dynamics of the Madden–Julian Oscillation (DYNAMO) field experiment in late 2011. The analysis is performed using data from the sounding array complemented by additional observational datasets for surface turbulent fluxes and atmospheric radiative heating. The entire analysis is repeated using the ECMWF Interim Re-Analysis (ERA-Interim). The roles of surface turbulent fluxes, radiative heating, and advection are quantified for the two MJO events that occurred in October and November using the sounding data; a third event in December is also studied in the ERA-Interim data. These results are consistent with the view that the MJO’s moist static energy anomalies grow and are sustained to a significant extent by the radiative feedbacks associated with MJO water vapor and cloud anomalies and that propagation of the MJO is associated with advection of moist static energy. Both horizontal and vertical advection appear to play significant roles in the events studied here. Horizontal advection strongly moistens the atmosphere during the buildup to the active phase of the October event when the low-level winds switch from westerly to easterly. Horizontal advection strongly dries the atmosphere in the wake of the active phases of the November and December events as the westerlies associated with off-equatorial cyclonic gyres bring subtropical dry air into the convective region from the west and north. Vertical advection provides relative moistening ahead of the active phase and drying behind it, associated with an increase of the normalized gross moist stability.Atmospheric sciences, Remote sensing, Climate change, Applied mathematics, Madden-Julian oscillation, Atmospheric circulationahs129, sw2526Applied Physics and Applied MathematicsArticlesBifurcation of On-site and Off-site Solitary Waves of Discrete Nonlinear Schrödinger Type Equations
http://academiccommons.columbia.edu/catalog/ac:189391
Jenkinson, Michael Jameshttp://dx.doi.org/10.7916/D8J102F0Wed, 19 Aug 2015 00:00:00 +0000A feature of immeasurable interest in nonlinear systems is that of spatially localized traveling pulses, or solitary waves - states which persist indefinitely in time, focus energy, and facilitate its transfer. Furthermore, in many lattice systems, discreteness effects are important and play a key role in these dynamics. In this thesis, we construct the multiple families of solitary standing (time-periodic) waves of the discrete, focusing cubically nonlinear Schrödinger equation (DNLS). These states are related to the so-called Peierls-Nabarro energy barrier, which refers to the energy difference between these distinct states and is thought to be responsible for the absence of indefinitely traveling, non-deforming solitary (spatially localized) waves of arbitrary velocity in many (non-dissipative) discrete systems. Instead, one observes that traveling waves of many discrete equations radiate energy and deform until they eventually cease to propagate and settle to a stationary time-periodic standing wave centered at a vertex. We address two specific cases of DNLS: (1) nearest-neighbor coupling on a cubic lattice in dimensions d = 1,2,3, and (2) long-range site coupling in dimension d = 1. These states are obtained via a bifurcation analysis about the continuum nonlinear Schrödinger equation (NLS) limit, with respect to a natural small parameter. Depending on the spatial dimension, these may be vertex-, bond-, cell-, or face-centered. In the first case of nearest-neighbor coupling, we construct an explicit asymptotic expansion. In the second case of one-dimensional long-range coupling when the decay of the site coupling with respect to distance is sufficiently slow, the continuum limiting NLS equation has Laplacian of fractional power. Finally, we show that the energy difference among distinct states of the same frequency is exponentially small with respect to the small parameter beyond all polynomial orders. This provides a rigorous bound for the Peierls-Nabarro barrier.Applied mathematics, Mathematics, Physicsmjj2122Applied Physics and Applied MathematicsDissertationsAnisotropic inverse problems with internal measurements
http://academiccommons.columbia.edu/catalog/ac:187031
Guo, Chenxihttp://dx.doi.org/10.7916/D88914XRMon, 11 May 2015 15:32:01 +0000This thesis concerns the hybrid inverse problem of reconstructing a tensor-valued conductivity from knowledge of internal measurements. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.
In the first part of the thesis, we investigate the reconstruction of the anisotropic conductivity in a second-order elliptic partial differential equation, from knowledge of internal current densities. We show that the unknown coefficient can be uniquely and stably reconstructed via explicit inversion formulas with a loss of one derivative compared to errors in the measurement. This improves the resolution of quantitative reconstructions in Calderón's problem(i.e. reconstruction problems from knowledge of boundary measurements). We then extend the problem to the full anisotropic Maxwell system and show that the complex-valued anisotropic admittivity can be uniquely reconstructed from knowledge of several internal magnetic fields. We also proved a unique continuation property and Runge approximation property for an anisotropic Maxwell system.
In the second part, we performed some numerical experiments to demonstrate the computational feasibility of the reconstruction algorithms and assess their robustness to noisy measurements.Applied mathematicscg2597Applied Physics and Applied MathematicsDissertationsBifurcation of localized eigenstates of perturbed periodic Schrödinger operators
http://academiccommons.columbia.edu/catalog/ac:182973
Vukicevic, Ivahttp://dx.doi.org/10.7916/D88C9V2ZThu, 05 Feb 2015 18:20:43 +0000A spatially localized initial condition for an energy-conserving wave equation with periodic coefficients disperses (spatially spreads) and decays as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of spatially localized perturbations in a periodic medium leads to ``defect modes'', states in which the wave is spatially localized and periodic in time. These modes are associated with eigenvalues which bifurcate from the continuous spectrum induced by the perturbation.
This thesis investigates specific families of perturbations of one-dimensional periodic Schrödinger operators and studies the resulting bifurcating eigenvalues from the unperturbed continuous spectrum. For Q(x) a real-valued periodic function, the Schrödinger operator H_Q = -∂_x^2 + Q(x) has a continuous spectrum equal to the union of closed intervals, called spectral bands, separated by open spectral gaps. We find that upon the introduction of a bounded, ``small'', and sufficiently decaying perturbation W(x), the spectrum of H_{Q+W} has discrete eigenvalues (with corresponding eigenstates which are exponentially decaying in |x|) which lie in the open spectral gaps of H_Q.
Our analysis covers two large classes of perturbations W(x): 1. W(x) = λ V(x), 0<λ ≪ 1, and V(x) sufficiently rapidly decaying as x → ± ∞; 2. W(x) = q(x, x/ε), 0<ε ≪ 1, where x ⟼ q(x,y) is spatially localized, q(x,y+1) = q(x,y) for x ∈ ℝ, and y ⟼ q(x,y) has mean zero.
In Case 1. W(x) corresponds to a small and localized absolute change in the medium's material properties. In Case 2. W(x) corresponds to a high-contrast microstructure. Q(x) + W(x) may be pointwise very large, but on average it is a small perturbation of Q(x).Applied mathematics, Mathematicsiv2143Applied Physics and Applied MathematicsDissertationsCloud Vertical Distribution across Warm and Cold Fronts in CloudSat–CALIPSO Data and a General Circulation Model
http://academiccommons.columbia.edu/catalog/ac:183326
Naud, Catherine M.; Del Genio, Anthony D.; Bauer, Mike; Kovari, Williamhttp://dx.doi.org/10.7916/D8BC3XC5Thu, 05 Feb 2015 00:00:00 +0000Cloud vertical distributions across extratropical warm and cold fronts are obtained using two consecutive winters of CloudSat–Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) observations and National Centers for Environmental Prediction reanalysis atmospheric state parameters over the Northern and Southern Hemisphere oceans (30°–70°N/S) between November 2006 and September 2008.These distributions generally resemble those from the original model introduced by the Bergen School in the 1920s, with the following exceptions: 1) substantial low cloudiness, which is present behind and ahead of the warm and cold fronts; 2) ubiquitous high cloudiness, some of it very thin, throughout the warm-frontal region; and 3) upright convective cloudiness near and behind some warm fronts. One winter of GISS general circulation model simulations of Northern and Southern Hemisphere warm and cold fronts at 2° x 2.5° x 32 levels resolution gives similar cloud distributions but with much lower cloud fraction, a shallower depth of cloudiness, and a shorter extent of tilted warm-frontal cloud cover on the cold air side of the surface frontal position. A close examination of the relationship between the cloudiness and relative humidity fields indicates that water vapor is not lifted enough in modeled midlatitude cyclones and this is related to weak vertical velocities in the model. The model also produces too little cloudiness for a given value of vertical velocity or relative humidity. For global climate models run at scales coarser than tens of kilometers, the authors suggest that the current underestimate of modeled cloud cover in the storm track regions, and in particular the 50°–60°S band of the Southern Oceans, could be reduced with the implementation of a slantwise convection parameterization.Climate change, Environmental studies, Applied mathematicscn2140, add1, mpb20, wk14Applied Physics and Applied Mathematics, Center for Climate Systems ResearchArticlesClustering on the Main Diagonal in Mobility Matrices
http://academiccommons.columbia.edu/catalog/ac:178652
Singer, Burton; Spilerman, Seymourhttp://dx.doi.org/10.7916/D8X63KJTWed, 15 Oct 2014 00:00:00 +0000In this paper we compare Markov, semi-Markov, and heterogeneity formulations. We demonstrate that the familiar phenomenon of clustering on the main diagonal is an inherent consequence of the structure of certain model types. Further, underprediction of the main diagonal elements should be associated with other kinds of stochastic models, beyond Markovian, that are also applicable to social processesSociology, Applied mathematicsss50SociologyArticlesMathematical Modeling of Insider Trading
http://academiccommons.columbia.edu/catalog/ac:178871
Bilina Falafala, Roselinehttp://dx.doi.org/10.7916/D89W0D33Mon, 13 Oct 2014 00:00:00 +0000In this thesis, we study insider trading and consider a financial market and an enlarged financial market whose sets of information are respectively represented by the filtrations F and G. The filtration G is obtained by initially expanding the filtration F. We also consider that we have a finite trading horizon. First, we show that under certain conditions the enlarged market satisfies no free lunch with vanishing risk (NFLVR) locally and therefore satisfies no arbitrage with respect to admissible simple predictable trading strategies. In addition, we generalize the structure of all the G local martingale deflators and find sufficient conditions under which the enlarged market satisfies NFLVR. We apply our results to some recent examples of insider trading that have appeared in newspapers and by doing so, show the limitations of some previous works that have studied the stability of the NFLVR property under an initial expansion. \newline Second, assuming the enlarged market satisfies NFLVR and markets are incomplete, we define a notion of risk and compare the risk of a market or liquidity trader to the risk of an insider trader. We prove that the risk of an insider is smaller than the risk of a market/liquidity trader under some sufficient conditions that involve their respective trading strategies. We find a relationship between the trading strategies of a market trader and of an insider when the risk neutral measure of the market is used. If an insider trades using the market risk neutral measure and not her own, then her trading strategy should involve not only the stock but also the volatility of the stock. \newline Finally, assuming that the enlarged market satisfies NFLVR locally, we provide a way for an insider to price her financial claims. We also define a new type of process that we call a quasi-local martingale and prove that the stock price process under local NFLVR is one such process.Applied mathematics, FinanceStatisticsDissertationsAdvancements for three-dimensional remote sensing of the atmosphere
http://academiccommons.columbia.edu/catalog/ac:178170
Martin, William George Kuleszhttp://dx.doi.org/10.7916/D8WM1BZCTue, 23 Sep 2014 00:00:00 +0000Climate modeling efforts depend on remote sensing observations of clouds and aerosols in the atmosphere. This dissertation presents a foundation for using three-dimensional (3D) remote sensing techniques to retrieve cloud and aerosol properties in complex cloud fields. The initial research was aimed at establishing a set of single-scattering properties that could be used in subsequent 3D remote sensing applications. A theoretical stability analysis was used to evaluate what information about the particulate scattering material could be determined from in situ radiance and polarization measurements, and particle size and refractive index were retrieved from synthetic measurements with noise levels comparable to those of existing laboratory instruments. Subsequent research focused on the techniques necessary to retrieve 3D atmosphere and surface properties from images taken by an airborne or space-borne instrument. With the goal of using 3D retrieval methods to extend monitoring capabilities to regions with broken cloud fields, we formulated an efficient procedure for using codes that solve the 3D vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties to fit multi-angle/multi-pixel polarimetric measurements of the atmosphere. Taken together, these two bodies of work contribute to ongoing research which focuses on developing new methods for retrieving aerosols in complex 3D cloud fields, and may extend monitoring capabilities to these currently unresolved scenes.Applied mathematics, Atmospheric scienceswgm2111Applied Physics and Applied MathematicsDissertationsStatistical method for revealing form-function relations in biological networks.
http://academiccommons.columbia.edu/catalog/ac:177513
Mugler, Andrew; Grinshpun, Boris; Franks, Riley; Wiggins, Chris H.http://dx.doi.org/10.7916/D8JQ0ZJ0Sat, 20 Sep 2014 00:00:00 +0000Over the past decade, a number of researchers in systems biology have sought to relate the function of biological systems to their network-level descriptions—lists of the most important players and the pairwise interactions between them. Both for large networks (in which statistical analysis is often framed in terms of the abundance of repeated small subgraphs) and for small networks which can be analyzed in greater detail (or even synthesized in vivo and subjected to experiment), revealing the relationship between the topology of small subgraphs and their biological function has been a central goal. We here seek to pose this revelation as a statistical task, illustrated using a particular setup which has been constructed experimentally and for which parameterized models of transcriptional regulation have been studied extensively. The question “how does function follow form” is here mathematized by identifying which topological attributes correlate with the diverse possible information-processing tasks which a transcriptional regulatory network can realize. The resulting method reveals one form-function relationship which had earlier been predicted based on analytic results, and reveals a second for which we can provide an analytic interpretation. Resulting source code is distributed via http://formfunction.sourceforge.net.Applied mathematicsbg2178, chw2Applied Physics and Applied Mathematics, Cellular, Molecular, Structural, and Genetic StudiesArticlesA 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 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 MathematicsArticlesMultiple 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 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 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 MathematicsArticlesLearning ‘‘graph-mer’’ Motifs that Predict Gene Expression Trajectories in Development
http://academiccommons.columbia.edu/catalog/ac:177504
Li, Xuejing; Panea, Casandra; Wiggins, Chris H.; 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 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 MathematicsArticlesTime-dependent information transmission in a model regulatory circuit
http://academiccommons.columbia.edu/catalog/ac:177464
Mancini, F.; 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 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 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 MathematicsArticlesStudies in Stochastic Networks: Efficient Monte-Carlo Methods, Modeling and Asymptotic Analysis
http://academiccommons.columbia.edu/catalog/ac:177127
Dong, Jinghttp://dx.doi.org/10.7916/D8X63K4FTue, 12 Aug 2014 00:00:00 +0000This dissertation contains two parts. The first part develops a series of bias reduction techniques for: point processes on stable unbounded regions, steady-state distribution of infinite server queues, steady-state distribution of multi-server loss queues and loss networks and sample path of stochastic differential equations. These techniques can be applied for efficient performance evaluation and optimization of the corresponding stochastic models. We perform detailed running time analysis under heavy traffic of the perfect sampling algorithms for infinite server queues and multi-server loss queues and prove that the algorithms achieve nearly optimal order of complexity. The second part aims to model and analyze the load-dependent slowdown effect in service systems. One important phenomenon we observe in such systems is bi-stability, where the system alternates randomly between two performance regions. We conduct heavy traffic asymptotic analysis of system dynamics and provide operational solutions to avoid the bad performance region.Operations research, Applied mathematicsjd2736Industrial Engineering and Operations ResearchDissertationsBayesian Hierarchical Rule Modeling for Predicting Medical Conditions
http://academiccommons.columbia.edu/catalog/ac:173882
McCormick, Tyler H.; Rudin, Cynthia; Madigan, David B.http://dx.doi.org/10.7916/D8V69GP1Wed, 14 May 2014 00:00:00 +0000We propose a statistical modeling technique, called the Hierarchical Association Rule Model (HARM), that predicts a patient’s possible future medical conditions given the patient’s current and past history of reported conditions. The core of our technique is a Bayesian hierarchical model for selecting predictive association rules (such as “condition 1 and condition 2 → condition 3”) from a large set of candidate rules. Because this method “borrows strength” using the conditions of many similar patients, it is able to provide predictions specialized to any given patient, even when little information about the patient’s history of conditions is available.Applied mathematics, Statistics, Medicinedm2418StatisticsArticlesLocation Estimation in Wireless Networks: A Bayesian Approach
http://academiccommons.columbia.edu/catalog/ac:173820
Madigan, David B.; Ju, Wen-Hua; Krishnan, P.; Krishnakumar, A. S. ; Zorych, Ivanhttp://dx.doi.org/10.7916/D82V2D74Tue, 13 May 2014 00:00:00 +0000We present a Bayesian hierarchical model for indoor location estimation in wireless networks. We demonstrate that out model achieves accuracy that is similar to other published models and algorithms. By harnessing prior knowledge, our model drastically reduces the requirement for training data as compared with existing approaches.Mathematics, Statistics, Applied mathematicsdm2418StatisticsArticlesPortfolio optimization with transaction costs and capital gain taxes
http://academiccommons.columbia.edu/catalog/ac:197100
Shen, Weiweihttp://dx.doi.org/10.7916/D8PK0D76Fri, 11 Apr 2014 16:41:52 +0000This thesis is concerned with a new computational study of optimal investment decisions with proportional transaction costs or capital gain taxes over multiple periods. The decisions are studied for investors who have access to a risk-free asset and multiple risky assets to maximize the expected utility of terminal wealth. The risky asset returns are modeled by a discrete-time multivariate geometric Brownian motion. As in the model in Davis and Norman (1990) and Lynch and Tan (2010), the transaction cost is modeled to be proportional to the amount of transferred wealth. As in the model in Dammon et al. (2001) and Dammon et al. (2004), the taxation rule is linear, uses the weighted average tax basis price, and allows an immediate tax credit for a capital loss. For the transaction costs problem, we compute both lower and upper bounds for optimal solutions. We propose three trading strategies to obtain the lower bounds: the hyper-sphere strategy (termed HS); the hyper-cube strategy (termed HC); and the value function optimization strategy (termed VF). The first two strategies parameterize the associated no-trading region by a hyper-sphere and a hyper-cube, respectively. The third strategy relies on approximate value functions used in an approximate dynamic programming algorithm. In order to examine their quality, we compute the upper bounds by a modified gradient-based duality method (termed MG). We apply the new methods across various parameter sets and compare their results with those from the methods in Brown and Smith (2011). We are able to numerically solve problems up to the size of 20 risky assets and a 40-year-long horizon. Compared with their methods, the three novel lower bound methods can achieve higher utilities. HS and HC are about one order of magnitude faster in computation times. The upper bounds from MG are tighter in various examples. The new duality gap is ten times narrower than the one in Brown and Smith (2011) in the best case. In addition, I illustrate how the no-trading region deforms when it reaches the borrowing constraint boundary in state space. To the best of our knowledge, this is the first study of the deformation in no-trading region shape resulted from the borrowing constraint. In particular, we demonstrate how the rectangular no-trading region generated in uncorrelated risky asset cases (see, e.g., Lynch and Tan, 2010; Goodman and Ostrov, 2010) transforms into a non-convex region due to the binding of the constraint.For the capital gain taxes problem, we allow wash sales and rule out "shorting against the box" by imposing nonnegativity on portfolio positions. In order to produce accurate results, we sample the risky asset returns from its continuous distribution directly, leading to a dynamic program with continuous decision and state spaces. We provide ingredients of effective error control in an approximate dynamic programming solution method. Accordingly, the relative numerical error in approximating value functions by a polynomial basis function is about 10E-5 measured by the l1 norm and about 10E-10 by the l2 norm. Through highly accurate numerical solutions and transformed state variables, we are able to explain the optimal trades through an associated no-trading region. We numerically show in the new state space the no-trading region has a similar shape and parameter sensitivity to that of the transaction costs problem in Muthuraman and Kumar (2006) and Lynch and Tan (2010). Our computational results elucidate the impact on the no-trading region from volatilities, tax rates, risk aversion of investors, and correlations among risky assets. To the best of our knowledge, this is the first time showing no-trading region of the capital gain taxes problem has such similar traits to that of the transaction costs problem. We also compute lower and upper bounds for the problem. To obtain the lower bounds we propose five novel trading strategies: the value function optimization (VF) strategy from approximate dynamic programming; the myopic optimization and the rolling buy-and-hold heuristic strategies (MO and RBH); and the realized Merton's and hyper-cube strategies (RM and HC) from policy approximation. In order to examine their performance, we develop two upper bound methods (VUB and GUB) based on the duality technique in Brown et al. (2009) and Brown and Smith (2011). Across various sets of parameters, duality gaps between lower and upper bounds are smaller than 3% in most examples. We are able to solve the problem up to the size of 20 risky assets and a 30-year-long horizon.Applied mathematicsws2215Applied Physics and Applied Mathematics, BusinessDissertationsA Spherical Chicken
http://academiccommons.columbia.edu/catalog/ac:172754
Stellman, Steven D.http://dx.doi.org/10.7916/D80863C7Fri, 04 Apr 2014 00:00:00 +0000Letter to the editor, published in Science (magazine).Biochemistry, Applied mathematicssds91EpidemiologyArticlesThermodyanic Properties of Poly(trans 1,4-butadiene) Crystals. Relationship to Molecular Structure
http://academiccommons.columbia.edu/catalog/ac:172751
Stellman, Jeanne M.; Woodward, Arthur E.; Stellman, Steven D.http://dx.doi.org/10.7916/D87S7KV1Fri, 04 Apr 2014 00:00:00 +0000Heat capacity measurements of melt crystallized poly( trans-1,4-butadiene) (PTBD) were carried out in the 50-130" region and the entropy change from 73" to the melting point, 139", was calculated. A value of the entropy change obtained using the rotational isomeric state approximation is found to underestimate the experimental entropy change. Theoretical energy calculations were carried out using empirical potential energy functions for a single PTBD chain, a unit cell and a lattice of cells. Minimization of the lattice energy with respect to two of the monoclinic cell constants for the low-temperature crystal form gave results in good agreement with X-ray diffraction data. The energy of transition from the low-temperature form was calculated and a theoretical heat capacity curve was obtained.Polymer chemistry, Applied mathematics, Chemistryjms13, sds91Epidemiology, Health Policy and ManagementArticlesComputer Simulation of Polymer Conformation. II. Distribution Function for Polymers with Excluded Volume
http://academiccommons.columbia.edu/catalog/ac:172739
Stellman, Steven D.; Gans, Paul J.http://dx.doi.org/10.7916/D8SQ8XGWFri, 04 Apr 2014 00:00:00 +0000Numerical distributions of end-to-end distances were generated by a Monte Carlo method for hard-sphere off-lattice polymers of length N = 20, 40, 60, 80, 98, and 298 atoms. Comparison by xz tests against five recently proposed theoretical distribution functions showed that for N = 80 and N = 98, the data could be described, with 95% confidence, by the equation f(r) = exp[ -(ar2 + br + c)], where a and b are fitted parameters and c is a normalization constant. For N = 298, limitations of sample size lead to lower confidence limits (about 80%), but good fit. The above equation, and not its gaussian counterpart exp( -cr^2), is probably the limiting distribution function. The function accurately predicts the 1st through 12th observed moments at all chain lengths.Polymer chemistry, Applied mathematicssds91EpidemiologyArticlesEfficient Computer Simulation of Polymer Conformation. I. Geometric Properties of the Hard-Sphere Model
http://academiccommons.columbia.edu/catalog/ac:172736
Stellman, Steven D.; Gans, Paul J.http://dx.doi.org/10.7916/D8N58JF4Fri, 04 Apr 2014 00:00:00 +0000A system of efficient computer programs has been developed for simulating the conformations of macromolecules. The conformation of an individual polymer is defined as a point in conformation space, whose mutually orthogonal axes represent the successive dihedral angles of the backbone chain. The statistical-mechanical average of any property is obtained as the usual configuration integral over this space. A Monte Carlo method for estimating averages is used because of the impossibility of direct numerical integration. Monte Carlo corresponds to the execution of a Markoffian random walk of a representative point through the conformation space. Unlike many previous Monte Carlo studies of polymers, which sample conformation space indiscriminately, importance sampling increases efficiency because selection of new polymers is biased to reflect their Boltzmann probabilities in the canonical ensemble, leading to reduction of sampling variance and hence to greater accuracy! in given computing time. The simulation is illustrated in detail. Overall running time is proportional to n^(5/4), where n is the chain length. Results are presented for a hard-sphere linear polymer of n atoms, with free dihedral rotation, with n = 20-298. The fraction of polymers accepted in the importance sampling scheme, fA, is fit to a Fisher-Sykes attrition relation, giving an effective attrition constant of zero. fA is itself an upper bound to the partition function, Q, relative to the unrestricted walk. The mean-squared end-to-end distance and radius of gyration exhibit the expected exponential dependence, but with exponent for the radius of gyration significantly greater than that of the end-to-end distance. The 90% confidence limits calculated for both exponents did not include either 6/5 or 4/3, the lattice and zero-order perturbation values, respectively. A self-correcting scheme for generating coordinates free of roundoff error is given in an Appendix.Applied mathematics, Polymer chemistrysds91EpidemiologyArticlesConformation of Guanosine Cytidine 3',5'-Monophosphate (GpC)
http://academiccommons.columbia.edu/catalog/ac:172748
Stellman, Steven D.; Hingerty, B.; Broyde, S. B.; Subramanian, E.; Sato, T.; Langridge, R.http://dx.doi.org/10.7916/D8CJ8BKRFri, 04 Apr 2014 00:00:00 +0000A brief communication of preliminary results of solution of dinucleoside phosphate GpC.Applied mathematics, Polymer chemistrysds91EpidemiologyArticlesComputer Graphics in the Solution of the Chain Deformation Problem
http://academiccommons.columbia.edu/catalog/ac:172760
Stellman, Steven D.http://dx.doi.org/10.7916/D8QR4V6PFri, 04 Apr 2014 00:00:00 +0000Go and Scheraga have given a general solution for determining the possible conformations of a section of a flexible polymeric chain between two fixed bonds (if the bonds coincide, the problem is equivalent to ring closure). Often, the fixed bonds are related by symmetry, such as that imposed by helicity or folding of antiparallel chains. A technique is shown for obtaining solutions using interactive computer graphics, whereby a plausible chain conformation or fold is approximated manually and completed by mathematical minimization. The procedure allows rapid, repetitive exploration of multiple solutions, and has many options for changing constraints. As an example, the backbone conformation of GpC, a dinucleoside phosphate, is used as a starting model, and gives upon minimization a set of backbone angles in the range of 11-fold helical RNA.Polymer chemistry, Applied mathematicssds91EpidemiologyArticlesNonlinear Data Assimilation: Towards a Prediction of the Solar Cycle
http://academiccommons.columbia.edu/catalog/ac:171928
Svedin, Andreashttp://dx.doi.org/10.7916/D8TH8JR5Thu, 20 Mar 2014 00:00:00 +0000The solar cycle is the cyclic variation of solar activity, with a span of 9-14 years. The prediction of the solar cycle is an important and unsolved problem with implications for communications, aviation and other aspects of our high-tech society. Our interest is model-based prediction, and we present a self-consistent procedure for parameter estimation and model state estimation, even when only one of several model variables can be observed. Data assimilation is the art of comparing, combining and transferring observed data into a mathematical model or computer simulation. We use the 3DVAR methodology, based on the notion of least squares, to present an implementation of a traditional data assimilation. Using the Shadowing Filter - a recently developed method for nonlinear data assimilation - we outline a path towards model based prediction of the solar cycle. To achieve this end we solve a number of methodological challenges related to unobserved variables. We also provide a new framework for interpretation that can guide future predictions of the Sun and other astrophysical objects.Astronomy, Astrophysics, Applied mathematicsaos2112AstronomyDissertationsLie-group interpolation and variational recovery for internal variables
http://academiccommons.columbia.edu/catalog/ac:171906
Mota, Alejandro; Sun, WaiChing; Ostien, Jakob T.; Long, Kevin N.; Foulk, James W.http://dx.doi.org/10.7916/D8H70CW9Wed, 19 Mar 2014 00:00:00 +0000We propose a variational procedure for the recovery of internal variables, in effect extending them from integration points to the entire domain. The objective is to perform the recovery with minimum error and at the same time guarantee that the internal variables remain in their admissible spaces. The minimization of the error is achieved by a three-field finite element formulation. The fields in the formulation are the deformation mapping, the target or mapped internal variables and a Lagrange multiplier that enforces the equality between the source and target internal variables. This formulation leads to an L2 projection that minimizes the distance between the source and target internal variables as measured in the L2 norm of the internal variable space. To ensure that the target internal variables remain in their original space, their interpolation is performed by recourse to Lie groups, which allows for direct polynomial interpolation of the corresponding Lie algebras b! y means of the logarithmic map. Once the Lie algebras are interpolated, the mapped variables are recovered by the exponential map, thus guaranteeing that they remain in the appropriate space.Theoretical mathematics, Applied mathematicsws2414Civil Engineering and Engineering MechanicsArticlesInfrastructure Scaling and Pricing
http://academiccommons.columbia.edu/catalog/ac:171000
Gocmen, Fikret Canerhttp://dx.doi.org/10.7916/D8SQ8XFFTue, 18 Feb 2014 00:00:00 +0000Infrastructure systems play a crucial role in our daily lives. They include, but are not limited to, the highways we take while we commute to work, the stadiums we go to watch games, and the power plants that provide the electricity we consume in our homes. In this thesis we study infrastructure systems from several different perspectives with a focus on pricing and scalability. The pricing aspect of our research focuses on two industries: toll roads and sports events. Afterwards, we analyze the potential impact of small modular infrastructure on a wide variety of industries. We start by analyzing the problem of determining the tolls that maximize revenue for a managed lane operator -- that is, an operator who can charge a toll for the use of some lanes on a highway while a number of parallel lanes remain free to use. Managing toll lanes for profit is becoming increasingly common as private contractors agree to build additional lane capacity in return for the opportunity to retain toll revenue. We start by modeling the lanes as queues and show that the dynamic revenue-maximizing toll is always greater than or equal to the myopic toll that maximizes expected revenue from each arriving vehicle. Numerical examples show that a dynamic revenue-maximizing toll scheme can generate significantly more expected revenue than either a myopic or a static toll scheme. An important implication is that the revenue-maximizing fee does not only depend on the current state, but also on anticipated future arrivals. We discuss the managerial implications and present several numerical examples. Next, we relax the queueing assumption and model traffic propagation on a highway realistically by using simulation. We devise a framework that can be used to obtain revenue maximizing tolls in such a context. We calibrate our framework by using data from the SR-91 Highway in Orange County, CA and explore different tolling schemes. Our numerical experiments suggest that simple dynamic tolling mechanisms can lead to substantial revenue improvements over myopic and time-of-use tolling policies. In the third part, we analyze the revenue management of consumer options for tournaments. Sporting event managers typically only offer advance tickets which guarantee a seat at a future sporting event in return for an upfront payment. Some event managers and ticket resellers have started to offer call options under which a customer can pay a small amount now for the guaranteed option to attend a future sporting event by paying an additional amount later. We consider the case of tournament options where the event manager sells team-specific options for a tournament final, such as the Super Bowl, before the finalists are determined. These options guarantee a final game ticket to the bearer if his team advances to the finals. We develop an approach by which an event manager can determine the revenue maximizing prices and amounts of advance tickets and options to sell for a tournament final. Afterwards, for a specific tournament structure we show that offering options is guaranteed to increase expected revenue for the event. We also establish bounds for the revenue improvement and show that introducing options can increase social welfare. We conclude by presenting a numerical application of our approach. Finally, we argue that advances made in automation, communication and manufacturing portend a dramatic reversal of the ``bigger is better'' approach to cost reductions prevalent in many basic infrastructure industries, e.g. transportation, electric power generation and raw material processing. We show that the traditional reductions in capital costs achieved by scaling up in size are generally matched by learning effects in the mass-production process when scaling up in numbers instead. In addition, using the U.S. electricity generation sector as a case study, we argue that the primary operating cost advantage of large unit scale is reduced labor, which can be eliminated by employing low-cost automation technologies. Finally, we argue that locational, operational and financial flexibilities that accompany smaller unit scale can reduce investment and operating costs even further. All these factors combined argue that with current technology, economies of numbers may well dominate economies of unit scale.Business, Operations research, Applied mathematicsBusinessDissertationsEfficient Computation of Polymer Conformation Energy
http://academiccommons.columbia.edu/catalog/ac:171438
Stellman, Steven D.; Froimowitz, Mark; Gans, Paul J.http://dx.doi.org/10.7916/D8W95757Fri, 14 Feb 2014 00:00:00 +0000Calculation of intramolecular energy of a polymer due to interactions of non-bonded atoms can be speeded up using simple geometric inequalities. This "zippering" method reduces time dependence of computation from n^2 to n^1.25, where n is the chain length. This technique is especially useful in applications of the pivot algorithm.Epidemiology, Applied mathematicssds91EpidemiologyArticlesToward an Understanding of Vertical Momentum Transports in Cloud-System-Resolving Model Simulations of Multiscale Tropical Convection
http://academiccommons.columbia.edu/catalog/ac:171435
Shaw, Tiffany A.; Lane, Todd P.http://dx.doi.org/10.7916/D80Z719SThu, 13 Feb 2014 00:00:00 +0000This study examines the characteristics of convective momentum transport (CMT) and gravity wave momentum transport (GWMT) in two-dimensional cloud-system-resolving model simulations, including the relationships between the two transports. A linear group velocity criterion is shown to objectively separate CMT and GWMT. The GWMT contribution is mostly consistent with upward-propagating gravity waves and is present in the troposphere and the stratosphere. The CMT contribution forms a large part of the residual (nonupward-propagating contribution) and dominates the fluxes in the troposphere. Additional analysis of the vertical sensible heat flux supports the physical interpretation of the two contributions, further isolating the effects of unstable convection from vertically propagating gravity waves. The role of transient and nonconservative (friction and diabatic heating) processes in generating momentum flux and their dependence on changes in convective organization was assessed using a pseudomomentum budget analysis. Nonconservative effects were found to dominate the transports; the GWMT contribution involved a diabatic source region in the troposphere and a dissipative sink region in the stratosphere. The CMT contribution was consistent with transport between the boundary layer and free troposphere via tilted convection. Transient buoyancy–vorticity correlations highlighted wave sources in the region of convective outflow and the boundary layer. These sources were akin to the previously described “mechanical oscillator” mechanism. Fluxes associated with this upper-level source were most sensitive to convective organization, highlighting the mechanism by which changes in organization are communicated to GWMT. The results elucidate important interactions between CMT and GWMT, adding further weight to suggestions that the two transports should be linked in parameterizations.Atmospheric sciences, Applied mathematicstas2163Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesIntraseasonal and Interannual Variability in North American Storm Tracks and Its Relationship to Equatorial Pacific Variability
http://academiccommons.columbia.edu/catalog/ac:174585
Grise, Kevin Michael; Son, Seok-Woo; Gyakum, John R.http://dx.doi.org/10.7916/D89021RVFri, 24 Jan 2014 00:00:00 +0000Extratropical cyclones play a principal role in wintertime precipitation and severe weather over North America. On average, the greatest number of cyclones track 1) from the lee of the Rocky Mountains eastward across the Great Lakes and 2) over the Gulf Stream along the eastern coastline of North America. However, the cyclone tracks are highly variable within individual winters and between winter seasons. In this study, the authors apply a Lagrangian tracking algorithm to examine variability in extratropical cyclone tracks over North America during winter. A series of methodological criteria is used to isolate cyclone development and decay regions and to account for the elevated topography over western North America. The results confirm the signatures of four climate phenomena in the intraseasonal and interannual variability in North American cyclone tracks: the North Atlantic Oscillation (NAO), the El Niño–Southern Oscillation (ENSO), the Pacific–North American pattern (PNA), and the Madden–Julian oscillation (MJO). Similar signatures are found using Eulerian bandpass-filtered eddy variances. Variability in the number of extratropical cyclones at most locations in North America is linked to fluctuations in Rossby wave trains extending from the central tropical Pacific Ocean. Only over the far northeastern United States and northeastern Canada is cyclone variability strongly linked to the NAO. The results suggest that Pacific sector variability (ENSO, PNA, and MJO) is a key contributor to intraseasonal and interannual variability in the frequency of extratropical cyclones at most locations across North America.Atmospheric sciences, Meteorology, Applied mathematicskmg2164Lamont-Doherty Earth ObservatoryArticlesOptimal Monetary and Fiscal Policy: A Linear-Quadratic Approach
http://academiccommons.columbia.edu/catalog/ac:167750
Benigno, Pierpaolo; Woodford, Michaelhttp://dx.doi.org/10.7916/D8H70CRHMon, 25 Nov 2013 00:00:00 +0000We propose an integrated treatment of the problems of optimal monetary and fiscal policy, for an economy in which prices are sticky (so that the supply-side effects of tax changes are more complex than in standard fiscal analyses) and the only available sources of government revenue are distorting taxes (so that the fiscal consequences of monetary policy must be considered alongside the usual stabilization objectives). Our linear-quadratic approach allows us to nest both conventional analyses of optimal monetary stabilization policy and analyses of optimal tax-smoothing as special cases of our more general framework. We show how a linear-quadratic policy problem can be derived which yields a correct linear approximation to the optimal policy rules from the point of view of the maximization of expected discounted utility in a dynamic stochastic general-equilibrium model. Finally, in addition to characterizing the optimal dynamic responses to shocks under an optimal policy, we derive policy rules through which the monetary and fiscal authorities may implement the optimal equilibrium. These take the form of optimal targeting rules, specifying an appropriate target criterion for each authority.Economics, Economic theory, Applied mathematicsmw2230EconomicsConferencesSelf-fulfilling Prophecies and the Business Cycle
http://academiccommons.columbia.edu/catalog/ac:167746
Farmer, Roger E. A.; Woodford, Michaelhttp://dx.doi.org/10.7916/D8N014G7Mon, 25 Nov 2013 00:00:00 +0000We demonstrate that multiple stationary rational-expectations equilibria exist in a version of Lucas's island economy. The existence of these equilibria follows from the fact that there is an indeterminate set of monetary equilibria in the two-period overlapping-generations model. We show how to construct stationary rational-expectations equilibria by randomizing over the set of nonstationary monetary equilibria. In some of our equilibria, a positively sloped Phillips curve exists even though our economy contains no signal-extraction problem as in the original Lucas paper. Our equilibria are indexed by beliefs and are examples of the existence of sunspot equilibria in which allocations may differ across states of nature for which preferences, technology, and endowments are identical. Our technique for constructing stationary sunspot equilibria should prove useful in a wide class of models in which an indeterminate stationary equilibrium exists.Economics, Economic theory, Applied mathematicsmw2230EconomicsArticlesThree Questions about Sunspot Equilibria as an Explanation of Economic Fluctuations
http://academiccommons.columbia.edu/catalog/ac:167419
Woodford, Michaelhttp://dx.doi.org/10.7916/D8DZ0671Thu, 21 Nov 2013 00:00:00 +0000It is by now well known that the sort of difference equations that characterize the equilibrium conditions of an infinite horizon competitive economy may have solutions in which the endogenous variables fluctuate in response to "sunspot" variables, that is, to random events that in fact have nothing to do with economic "fundamentals," and so do not directly affect the equilibrium conditions. It is possible to view such "sunspot equilibria" as a representation of an actual phenomenon economic fluctuations not caused by exogenous shocks to fundamentals, but rather by revisions of agents' expectations in response to some event, which revised expectations become self-fulfilling. Early discussions of such solutions sometimes suggested that a more rigorous derivation of the requirements for equilibrium might yield additional restrictions that would eliminate the sunspot solutions from the set of true equilibria. The demonstration by Karl Shell (1977), David Cass (1981), and Costas Azariadis (1981) that sunspot equilibria can exist in a rigorously formulated intertemporal equilibrium model, namely the overlapping generations model of Samuelson, has shown that this is not always the case. Nevertheless, many economists remain skeptical about the reasonableness of the sunspot hypothesis as a possible explanation of actual economic fluctuations, and for quite general reasons, independent of judgments about the empirical plausibility of any particular models. I discuss here three such general reasons for skepticism.Economics, Economic theory, Applied mathematicsmw2230EconomicsArticlesTwo-Layer Geostrophic Vortex Dynamics. Part 2. Alignment and Two-Layer V-States
http://academiccommons.columbia.edu/catalog/ac:167173
Polvani, Lorenzo M.http://dx.doi.org/10.7916/D8G44N6TTue, 19 Nov 2013 13:38:39 +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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 MathematicsArticlesLarge Scale Simulation of Spinodal Decomposition
http://academiccommons.columbia.edu/catalog/ac:189103
Zheng, Xianghttp://dx.doi.org/10.7916/D89W0DZ9Thu, 07 Nov 2013 00:00:00 +0000Spinodal decomposition is a process in which a system of binary mixture eventually evolves to the separation of two macroscopic phases. Such phase separation occurs in a thermodynamically unstable state. A number of binary mixture experiments have demonstrated the phenomenon of spinodal decomposition. Many models have been proposed to describe the evolution of the spinodal decomposition. The Cahn-Hilliard (CH) partial differential equation, which includes an order parameter and a free energy, and evolves to minimize the energy, has frequently been used as a phase field model. Due to random thermal fluctuations that are inevitable in physical systems, the CH equation might be unrealistic for the overall decomposition process. Experimental results demonstrate the existence of Brownian motion in the spinodal decomposition, which suggests that diffusion (deterministic contribution) and the noise (stochastic contribution) both have an essential influence on the rate of spinodal decomposition. Therefore, a stochastic process should be part of a realistic mathematical model of the overall decomposition process. In order to overcome the disadvantage that the CH equation ignores physically significant thermal fluctuation, the CH equation with a thermal fluctuation term has been proposed, where the thermal fluctuation is modeled by a time-space Brownian motion. The CH equation with the thermal fluctuation was first considered by Cook, so the extended CH equation is also known as the Cahn-Hilliard-Cook (CHC) equation. For studying the CHC equation, we are primarily interested in the properties of steady state, such as the energies, statistical moments, and morphology. This motivates our choices for the numerical frameworks for analyzing the CHC equation. The CHC equation is a stochastic partial differential equation involving a biharmonic form and a noise forcing term. When the potential term is a polynomial, the CHC equation is split into a lower order PDE system of two harmonic equations. The space is discretized by the standard finite element method. The evolution of the spinodal decomposition and the effect of the thermal fluctuation are studied in 2D. For obtaining numerical results of the CHC equation with a more realistic logarithmic potential efficiently, especially in 3D, a fully implicit, cell-centered, finite difference scheme in the original biharmonic form, and an adaptive time-stepping strategy are combined to discretize the space and time. The numerical scheme is verified by a comparison with an explicit scheme. At each time step, the parallel NKS algorithm is used to solve a nonlinear spatially discretized system. We discuss various numerical and computational challenges associated with the cell-centered finite difference-based, massively parallel implementation of this framework. We present steady state solutions of the CHC equation in 2D and, for the first time, in 3D. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We demonstrate that the thermal fluctuation is able to accelerate the spinodal decomposition process, and change the final steady morphology. We study the evolution of energies and statistical moments, from the initial stage to the steady state. Next, we study the CHC equation from the statistical perspective. A parallel domain decomposition method, based on the Wiener chaos expansion (WCE) and the Karhunen-Loeve expansion (KLE), is presented. Applying the two expansions to time-space white noise, we transform the CHC equation into a deterministic form. The main advantage of the Wiener chaos approach is that it separates deterministic and random effects, and factors the latter out of the primary stochastic partial differential equation effectively and rigorously. Therefore, the stochastic partial differential equation can be reduced to its propagator: a system of deterministic equations for the coefficients of the Wiener chaos expansion. Formulae for the expansion of high order nonlinear terms are presented, which involve the solutions of the propagator. Compared to the Monte Carlo (MC) method, the Wiener chaos approach does not require the generation of random numbers. The Karhunen-Loeve expansion is able to capture the principal component of the random field. A domain decomposition method is used to solve the equation system, which is discretized by a stabilized implicit cell-centered finite difference scheme. An NKS algorithm is applied to solve the nonlinear system of equations at each time step. The evolution of the spinodal decomposition and respective variances are demonstrated. Numerical results demonstrate that the parallel domain decomposition method scales well to a thousand processor cores. For short time, the Wiener chaos Karhunen-Loeve expansion (WCKLE) method is more efficient than the Monte Carlo simulation. We simulate the whole spinodal decomposition process by the Wiener chaos Karhunen-Loeve expansion Monte Carlo (WKCLE-MC) hybrid method, and obtain the distinctive separation stage for long time.Applied mathematicsApplied Physics and Applied MathematicsDissertationsForced-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 MathematicsArticlesA New Look at Stratospheric Sudden Warmings. Part II: Evaluation of Numerical Model Simulations
http://academiccommons.columbia.edu/catalog/ac:166924
Charlton, Andrew J.; Polvani, Lorenzo M.; Perlwitz, Judith; Sassi, Fabrizio; Manzini, Elisa; Pawson, Steven; Shibata, Kiyotaka; Nielsen, J. Eric; Rind, Davidhttp://hdl.handle.net/10022/AC:P:22113Mon, 04 Nov 2013 00:00:00 +0000The simulation of major midwinter stratospheric sudden warmings (SSWs) in six stratosphere-resolving general circulation models (GCMs) is examined. The GCMs are compared to a new climatology of SSWs, based on the dynamical characteristics of the events. First, the number, type, and temporal distribution of SSW events are evaluated. Most of the models show a lower frequency of SSW events than the climatology, which has a mean frequency of 6.0 SSWs per decade. Statistical tests show that three of the six models produce significantly fewer SSWs than the climatology, between 1.0 and 2.6 SSWs per decade. Second, four process-based diagnostics are calculated for all of the SSW events in each model. It is found that SSWs in the GCMs compare favorably with dynamical benchmarks for SSW established in the first part of the study. These results indicate that GCMs are capable of quite accurately simulating the dynamics required to produce SSWs, but with lower frequency than the climatology. Further dynamical diagnostics hint that, in at least one case, this is due to a lack of meridional heat flux in the lower stratosphere. Even though the SSWs simulated by most GCMs are dynamically realistic when compared to the NCEP–NCAR reanalysis, the reasons for the relative paucity of SSWs in GCMs remains an important and open question.Atmospheric sciences, Aeronomy, Applied mathematicsac2343, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesInternal 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 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 MathematicsArticlesA New Look at Stratospheric Sudden Warmings. Part I: Climatology and Modeling Benchmarks
http://academiccommons.columbia.edu/catalog/ac:166927
Charlton, Andrew J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22113Mon, 04 Nov 2013 00:00:00 +0000Stratospheric sudden warmings are the clearest and strongest manifestation of dynamical coupling in the stratosphere–troposphere system. While many sudden warmings have been individually documented in the literature, this study aims at constructing a comprehensive climatology: all major midwinter warming events are identified and classified, in both the NCEP–NCAR and 40-yr ECMWF Re-Analysis (ERA-40) datasets. To accomplish this a new, objective identification algorithm is developed. This algorithm identifies sudden warmings based on the zonal mean zonal wind at 60°N and 10 hPa, and classifies them into events that do and do not split the stratospheric polar vortex. Major midwinter stratospheric sudden warmings are found to occur with a frequency of approximately six events per decade, and 46% of warming events lead to a splitting of the stratospheric polar vortex. The dynamics of vortex splitting events is contrasted to that of events where the vortex is merely displaced off the pole. In the stratosphere, the two types of events are found to be dynamically distinct: vortex splitting events occur after a clear preconditioning of the polar vortex, and their influence on middle-stratospheric temperatures lasts for up to 20 days longer than vortex displacement events. In contrast, the influence of sudden warmings on the tropospheric state is found to be largely insensitive to the event type. Finally, a table of dynamical benchmarks for major stratospheric sudden warming events is compiled. These benchmarks are used in a companion study to evaluate current numerical model simulations of the stratosphere.Atmospheric sciences, Aeronomy, Applied mathematicsac2343, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesThe Effect of Lower Stratospheric Shear on Baroclinic Instability
http://academiccommons.columbia.edu/catalog/ac:166918
Wittman, Matthew A. H.; Charlton, Andrew J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22111Mon, 04 Nov 2013 00:00:00 +0000Using a hierarchy of models, and observations, the effect of vertical shear in the lower stratosphere on baroclinic instability in the tropospheric midlatitude jet is examined. It is found that increasing stratospheric shear increases the phase speed of growing baroclinic waves, increases the growth rate of modes with low synoptic wavenumbers, and decreases the growth rate of modes with higher wavenumbers. The meridional structure of the linear modes, and their acceleration of the zonal mean jet, changes with increasing stratospheric shear, but in a way that apparently contradicts the observed stratosphere–troposphere northern annular mode (NAM) connection. This contradiction is resolved at finite amplitude. In nonlinear life cycle experiments it is found that increasing stratospheric shear, without changing the jet structure in the troposphere, produces a transition from anticyclonic (LC1) to cyclonic (LC2) behavior at wavenumber 7. All life cycles with wavenumbers lower than 7 are LC1, and all with wavenumber greater than 7 are LC2. For the LC1 life cycles, the effect of increasing stratospheric shear is to increase the poleward displacement of the zonal mean jet by the eddies, which is consistent with the observed stratosphere–troposphere NAM connection. Finally, it is found that the connection between high stratospheric shear and high-tropospheric NAM is present by NCEP–NCAR reanalysis data.Atmospheric sciences, Aeronomy, Applied mathematicsmaw2006, ac2343, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesOn the Wavelength of the Rossby Waves Radiated by Tropical Cyclones
http://academiccommons.columbia.edu/catalog/ac:166903
Krouse, Kyle D.; Sobel, Adam H.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22106Mon, 04 Nov 2013 00:00:00 +0000The authors present a theory for the zonal wavelength of tropical depression–type disturbances, which occur as a result of Rossby wave radiation from a preexisting tropical cyclone (TC). In some cases, such disturbances undergo tropical cyclogenesis, resulting in a pair of tropical cyclones; the theory then predicts the zonal separation distance of such tropical cyclone pairs. Numerical experiments are presented in which a thermally forced vortex, superimposed on an initial state of rest, is moved at different velocities in a shallow-water model on a sphere. Vortices moving westward generate coherent wave trains to the east or southeast (depending on the amplitude of the vortex), resembling those in observations. The zonal wavelengths of these wave trains in each case are well described by the linear stationary solution in the frame comoving with the vortex. Vortices moving eastward or remaining stationary do not generate such trains, also consistent with linear theory, which admits no stationary solutions in such cases. It is hypothesized that the wavelengths of observed disturbances are set by the properties of the relevant stationary solution. The environmental flow velocity that determines this wavelength is not the translation velocity of the tropical cyclone, but the difference between the steering flow of the radiated Rossby waves and that of the TC. The authors argue that either horizontal or vertical shear in the environment of the TC can generate differences between these steering flows of the necessary magnitude and sign to generate the observed wavelengths.Atmospheric sciences, Applied mathematics, Meteorologykdk8, ahs129, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesInternal 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 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 MathematicsArticlesA Very Large, Spontaneous Stratospheric Sudden Warming in a Simple AGCM: A Prototype for the Southern Hemisphere Warming of 2002?
http://academiccommons.columbia.edu/catalog/ac:166956
Kushner, Paul J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22127Mon, 04 Nov 2013 00:00:00 +0000An exceptionally strong stratospheric sudden warming (SSW) that spontaneously occurs in a very simple stratosphere–troposphere AGCM is discussed. The model is a dry, hydrostatic, primitive equation model without planetary stationary waves. Transient baroclinic wave–wave interaction in the troposphere thus provides the only source of upward-propagating wave activity into the stratosphere. The model’s SSW is grossly similar to the Southern Hemisphere major SSW of 2002: it occurs after weaker warmings “precondition” the polar vortex for breaking, it involves a split of the polar vortex, and it has a downward-propagating signature. These similarities suggest that the Southern Hemisphere SSW of 2002 might itself have been caused by transient baroclinic wave–wave interaction. The simple model used for this study also provides some insight into how often such extreme events might occur. The frequency distribution of SSWs in the model has exponential, as opposed to Gaussian, tails. This suggests that very large amplitude SSWs, though rare, might occur with higher frequency than might be naively expected.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesStratosphere–Troposphere Coupling in a Relatively Simple AGCM: Impact of the Seasonal Cycle
http://academiccommons.columbia.edu/catalog/ac:166953
Kushner, Paul J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22126Mon, 04 Nov 2013 00:00:00 +0000The seasonal time dependence of the tropospheric circulation response to polar stratospheric cooling in a simple atmospheric general circulation model is investigated. When the model is run without a seasonal cycle, polar stratospheric cooling induces a positive annular-mode response in the troposphere that takes a remarkably long time—several hundred days—to fully equilibrate. One is thus led to ask whether the tropospheric response would survive in the presence of a seasonal cycle. When a seasonal cycle is introduced into the model stratosphere, the tropospheric response appears with a distinct time lag with respect to the stratospheric cooling, but, in the long-term mean, the pattern of the wind response is very similar to the one that results from stratospheric forcing in the absence of a seasonal cycle.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesStratospheric Influence on Baroclinic Lifecycles and its Connection to the Arctic Oscillation
http://academiccommons.columbia.edu/catalog/ac:166968
Wittman, Matthew A. H.; Polvani, Lorenzo M.; Charlton, Andrew J.; Scott, Richard K.http://hdl.handle.net/10022/AC:P:22131Mon, 04 Nov 2013 00:00:00 +0000Using an idealized primitive equation model, we investigate how stratospheric conditions alter the development of baroclinic instability in the troposphere. Starting from the lifecycle paradigm of Thorncroft et al., we consider the evolution of baroclinic lifecycles resulting from the addition of a stratospheric jet to the LC1 initial condition. We find that the addition of the stratospheric jet yields a net surface geopotential height anomaly that strongly resembles the Arctic Oscillation. With the additional modification of the tropospheric winds to resemble the high-AO climatology, the surface response is amplified by a factor 10 and, though dominated by the tropospheric changes, shows similar sensitivity to the stratospheric conditions.Atmospheric sciences, Applied mathematics, Aeronomymaw2006, lmp3, ac2343Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesOn the Meridional Structure of Annular Modes
http://academiccommons.columbia.edu/catalog/ac:166962
Wittman, Matthew A. H.; Charlton, Andrew J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22129Mon, 04 Nov 2013 00:00:00 +0000Using a simple stochastic model, the authors illustrate that the occurrence of a meridional dipole in the first empirical orthogonal function (EOF) of a time-dependent zonal jet is a simple consequence of the north–south excursion of the jet center, and this geometrical fact can be understood without appealing to fluid dynamical principles. From this it follows that one ought not, perhaps, be surprised at the fact that such dipoles, commonly referred to as the Arctic Oscillation (AO) or the Northern Annular Mode (NAM), have robustly been identified in many observational studies and appear to be ubiquitous in atmospheric models across a wide range of complexity.Atmospheric sciences, Aeronomy, Applied mathematicsmaw2006, ac2343, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesThe Hadley Circulation and the Weak Temperature Gradient Approximation
http://academiccommons.columbia.edu/catalog/ac:166986
Polvani, Lorenzo M.; Sobel, Adam H.http://hdl.handle.net/10022/AC:P:22137Mon, 04 Nov 2013 00:00:00 +0000The weak temperature gradient (WTG) approximation is applied to simple shallow-water models of the Hadley circulation. While it is difficult to formally justify the use of the WTG approximation for this problem, the derived WTG solutions are shown to agree well with numerical solutions of the full equations and to converge to the traditional angular momentum conserving (AMC) solutions in the inviscid limit. Heuristic arguments are given to explain this. The WTG method also provides semianalytical solutions in the case of nonvanishing viscosity, in contrast to the AMC solutions, which are strictly inviscid.Atmospheric sciences, Meteorology, Applied mathematicslmp3, ahs129Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesKelvin–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 MathematicsArticlesNumerically Converged Solutions of the Global Primitive Equations for Testing the Dynamical Core of Atmospheric GCMs
http://academiccommons.columbia.edu/catalog/ac:166983
Polvani, Lorenzo M.; Scott, R. K.; Thomas, S. J.http://hdl.handle.net/10022/AC:P:22136Mon, 04 Nov 2013 00:00:00 +0000Solutions of the dry, adiabatic, primitive equations are computed, for the first time, to numerical convergence. These solutions consist of the short-time evolution of a slightly perturbed, baroclinically unstable, midlatitude jet, initially similar to the archetypal LC1 case of Thorncroft et al. The solutions are computed with two distinct numerical schemes to demonstrate that they are not dependent on the method used to obtain them. These solutions are used to propose a new test case for dynamical cores of atmospheric general circulation models. Instantaneous horizontal and vertical cross sections of vorticity and vertical velocity after 12 days, together with tables of key diagnostic quantities derived from the new solutions, are offered as reproducible benchmarks. Unlike the Held and Suarez benchmark, the partial differential equations and the initial conditions are here completely specified, and the new test case requires only 12 days of integration, involves no spatial or temporal averaging, and does not call for physical parameterizations to be added to the dynamical core itself.Atmospheric sciences, Meteorology, Applied mathematicslmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesStratospheric 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 MathematicsArticlesStratosphere–Troposphere Coupling in a Relatively Simple AGCM: The Role of Eddies
http://academiccommons.columbia.edu/catalog/ac:166971
Kushner, Paul J.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22132Mon, 04 Nov 2013 00:00:00 +0000The extratropical circulation response to cooling of the polar-winter stratosphere in a simple AGCM is investigated. The AGCM is a dry hydrostatic primitive equation model with zonally symmetric boundary conditions and analytically specified physics. It is found that, as the polar-winter stratosphere is cooled, the tropospheric jet shifts poleward. This response projects almost entirely and positively (by convention) onto the AGCM's annular mode. At the same time, the vertical flux of wave activity from the troposphere to the stratosphere is reduced and the meridional flux of wave activity from high to low latitudes is increased. Thus, as the stratosphere is cooled, the stratospheric wave drag is reduced. In order to understand this response, the transient adjustment of the stratosphere–troposphere system is investigated using an ensemble of “switch on” stratospheric cooling runs of the AGCM. The response to the switch-on stratospheric cooling descends from the upper stratosphere into the troposphere on a time scale that matches simple downward-control theory estimates. The downward-control analysis is pursued with a zonally symmetric model that uses as input the thermal and eddy-driving terms from the eddying AGCM. With this model, the contributions to the response from the thermal and eddy-driving perturbations can be investigated separately, in the absence of eddy feedbacks. It is found that the stratospheric thermal perturbation, in the absence of such feedbacks, induces a response that is confined to the stratosphere. The stratospheric eddy-driving perturbation, on the other hand, induces a response that penetrates into the midtroposphere but does not account, in the zonally symmetric model, for the tropospheric jet shift. Furthermore, the tropospheric eddy-driving perturbation, in the zonally symmetric model, induces a strong upward response in the stratospheric winds. From these experiments and from additional experiments with the eddying AGCM, it is concluded that the stratospheric eddy-driving response induces a tropospheric response, but that the full circulation response results from a two-way coupling between the stratosphere and the troposphere.Atmospheric sciences, Aeronomy, Applied mathematicslmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesEquatorial Superrotation on Tidally Locked Exoplanets
http://academiccommons.columbia.edu/catalog/ac:166848
Showman, Adam P.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22087Fri, 01 Nov 2013 00:00:00 +0000The increasing richness of exoplanet observations has motivated a variety of three-dimensional (3D) atmospheric circulation models of these planets. Under strongly irradiated conditions, models of tidally locked, short-period planets (both hot Jupiters and terrestrial planets) tend to exhibit a circulation dominated by a fast eastward, or "superrotating," jet stream at the equator. When the radiative and advection timescales are comparable, this phenomenon can cause the hottest regions to be displaced eastward from the substellar point by tens of degrees longitude. Such an offset has been subsequently observed on HD 189733b, supporting the possibility of equatorial jets on short-period exoplanets. Despite its relevance, however, the dynamical mechanisms responsible for generating the equatorial superrotation in such models have not been identified. Here, we show that the equatorial jet results from the interaction of the mean flow with standing Rossby waves induced by the day-night thermal forcing. The strong longitudinal variations in radiative heating—namely intense dayside heating and nightside cooling—trigger the formation of standing, planetary-scale equatorial Rossby and Kelvin waves. The Rossby waves develop phase tilts that pump eastward momentum from high latitudes to the equator, thereby inducing equatorial superrotation. We present an analytic theory demonstrating this mechanism and explore its properties in a hierarchy of one-layer (shallow-water) calculations and fully 3D models. The wave-mean-flow interaction produces an equatorial jet whose latitudinal width is comparable to that of the Rossby waves, namely the equatorial Rossby deformation radius modified by radiative and frictional effects. For conditions typical of synchronously rotating hot Jupiters, this length is comparable to a planetary radius, explaining the broad scale of the equatorial jet obtained in most hot-Jupiter models. Our theory illuminates the dependence of the equatorial jet speed on forcing amplitude, strength of friction, and other parameters, as well as the conditions under which jets can form at all.Atmospheric sciences, Planetology, Applied mathematicslmp3Applied Physics and Applied Mathematics, Lamont-Doherty Earth ObservatoryArticlesThe Effect of Topography on Storm-Track Intensity in a Relatively Simple General Circulation Model
http://academiccommons.columbia.edu/catalog/ac:166878
Son, Seok-Woo; Polvani, Lorenzo M.; Ting, Mingfanghttp://hdl.handle.net/10022/AC:P:22098Fri, 01 Nov 2013 00:00:00 +0000The effect of topography on storm-track intensity is examined with a set of primitive equation model integrations. This effect is found to be crucially dependent on the latitudinal structure of the background flow impinging on the topography. If the background flow consists of a weak double jet, higher topography leads to an intensification of the storm track downstream of the topography, consistent with enhanced baroclinicity in that region. However, if the background flow consists of a strong single jet, topography weakens the storm track, despite the fact that the baroclinicity downstream of the topography is again enhanced. The different topographic impact results from the different wave packets in the two background flows. For a weak double-jet state, wave packets tend to radiate equatorward and storm-track eddies grow primarily at the expense of local baroclinicity. In contrast, for a strong single-jet state, wave packets persistently propagate in the zonal direction and storm tracks are affected not only by local baroclinicity but also by far-upstream disturbances via downstream development. It is the reduction of the latter by the topography that leads to weaker storm tracks in a strong single-jet state. The implications of these findings for Northern Hemisphere storm tracks are also discussed.Atmospheric sciences, Meteorology, Applied mathematicssws2112, lmp3, mt2204Ocean and Climate Physics, Applied Physics and Applied Mathematics, Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesAnnular Mode Time Scales in the Intergovernmental Panel on Climate Change Fourth Assessment Report Models
http://academiccommons.columbia.edu/catalog/ac:166896
Gerber, Edwin P.; Polvani, Lorenzo M.; Ancukiewicz, Damianhttp://hdl.handle.net/10022/AC:P:22103Fri, 01 Nov 2013 00:00:00 +0000The ability of climate models in the Intergovernmental Panel on Climate Change Fourth Assessment Report to capture the temporal structure of the annular modes is evaluated. The vertical structure and annual cycle of the variability is quantified by the e-folding time scale of the annular mode autocorrelation function. Models vaguely capture the qualitative features of the Northern and Southern Annular Modes: Northern Hemisphere time scales are shorter than those of the Southern Hemisphere and peak in boreal winter, while Southern Hemisphere time scales peak in austral spring and summer. Models, however, systematically overestimate the time scales, particularly in the Southern Hemisphere summer, where the multimodel ensemble average is twice that of reanalyses. Fluctuation-dissipation theory suggests that long time scales in models could be associated with increased sensitivity to anthropogenic forcing. Comparison of model pairs with similar forcings but different annular mode time scales provides a hint of a fluctuation-dissipation relationship.Atmospheric sciences, Applied mathematics, Climate changeepg2108, lmp3, da2260Applied Physics and Applied Mathematics, Engineering and Applied Science, Earth and Environmental SciencesArticlesEquatorial 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 MathematicsArticlesUnderstanding Hadley Cell Expansion versus Contraction: Insights from Simplified Models and Implications for Recent Observations
http://academiccommons.columbia.edu/catalog/ac:168459
Tandon, Neil F.; Gerber, Edwin P.; Sobel, Adam H.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22059Thu, 31 Oct 2013 00:00:00 +0000This study seeks a deeper understanding of the causes of Hadley Cell (HC) expansion, as projected under global warming, and HC contraction, as observed under El Niño. Using an idealized general circulation model, the authors show that a thermal forcing applied to a narrow region around the equator produces “El Niño–like” HC contraction, while a forcing with wider meridional extent produces “global warming–like” HC expansion. These circulation responses are sensitive primarily to the thermal forcing’s meridional structure and are less sensitive to its vertical structure. If the thermal forcing is confined to the midlatitudes, the amount of HC expansion is more than three times that of a forcing of comparable amplitude that is spread over the tropics. This finding may be relevant to recently observed trends of rapid tropical widening. The shift of the HC edge is explained using a very simple model in which the transformed Eulerian mean (TEM) circulation acts to diffuse heat meridionally. In this context, the HC edge is defined as the downward maximum of residual vertical velocity in the upper troposphere ϖmax *; this corresponds well with the conventional Eulerian definition of the HC edge. In response to a positive thermal forcing, there is anomalous diabatic cooling, and hence anomalous TEM descent, on the poleward flank of the thermal forcing. This causes the HC edge (ϖmax *) to shift toward the descending anomaly, so that a narrow forcing causes HC contraction and a wide forcing causes HC expansion.Atmospheric sciences, Applied mathematics, Climate changenft2104, ahs129, lmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesEl Niño, La Niña, and Stratospheric Sudden Warmings: A Reevaluation in Light of the Observational Record
http://academiccommons.columbia.edu/catalog/ac:166842
Butler, Amy H.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22085Thu, 31 Oct 2013 00:00:00 +0000Recent studies have suggested that El Niño-Southern Oscillation (ENSO) may have a considerable impact on Northern Hemisphere wintertime stratospheric conditions. Notably, during El Niño the stratosphere is warmer than during ENSO-neutral winters, and the polar vortex is weaker. Opposite-signed anomalies have been reported during La Niña, but are considerably smaller in amplitude than during El Niño. This has led to the perception that El Niño is able to substantially affect stratospheric conditions, but La Niña is of secondary importance. Here we revisit this issue, but focus on the extreme events that couple the troposphere to the stratosphere: major, mid-winter stratospheric sudden warmings (SSWs). We examine 53 years of reanalysis data and find, as expected, that SSWs are nearly twice as frequent during ENSO winters as during non-ENSO winters. Surprisingly, however, we also find that SSWs occur with equal probability during El Niño and La Niña winters. These findings corroborate the impact of ENSO on stratospheric variability, and highlight that both phases of ENSO are important in enhancing stratosphere-troposphere dynamical coupling via an increased frequency of SSWs.Atmospheric sciences, Meteorology, Applied mathematicslmp3Applied Physics and Applied Mathematics, Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesUncertainty in Climate Change Projections of the Hadley Circulation: The Role of Internal Variability
http://academiccommons.columbia.edu/catalog/ac:172651
Kang, Sarah M.; Deser, Clara; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22067Thu, 31 Oct 2013 00:00:00 +0000The uncertainty arising from internal climate variability in climate change projections of the Hadley circulation (HC) is presently unknown. In this paper it is quantified by analyzing a 40-member ensemble of integrations of the Community Climate System Model, version 3 (CCSM3), under the Special Report on Emissions Scenarios (SRES) A1B scenario over the period 2000–60. An additional set of 100-yr-long time-slice integrations with the atmospheric component of the same model [Community Atmosphere Model, version 3.0 (CAM3)] is also analyzed. Focusing on simple metrics of the HC—its strength, width, and height—three key results emerge from the analysis of the CCSM3 ensemble. First, the projected weakening of the HC is almost entirely confined to the Northern Hemisphere, and is stronger in winter than in summer. Second, the projected widening of the HC occurs only in the winter season but in both hemispheres. Third, the projected rise of the tropical tropopause occurs in both hemispheres and in all seasons and is, by far, the most robust of the three metrics. This paper shows further that uncertainty in future trends of the HC width is largely controlled by extratropical variability, while those of HC strength and height are associated primarily with tropical dynamics. Comparison of the CCSM3 and CAM3 integrations reveals that ocean–atmosphere coupling is the dominant source of uncertainty in future trends of HC strength and height and of the tropical mean meridional circulation in general. Finally, uncertainty in future trends of the hydrological cycle is largely captured by the uncertainty in future trends of the mean meridional circulation.Atmospheric sciences, Meteorology, Applied mathematicslmp3Applied Physics and Applied Mathematics, Earth and Environmental SciencesArticlesThe Antarctic Atmospheric Energy Budget. Part I: Climatology and Intraseasonal-to-Interannual Variability
http://academiccommons.columbia.edu/catalog/ac:172197
Previdi, Michael; Smith, Karen L.; Polvani, Lorenzo M.http://hdl.handle.net/10022/AC:P:22064Thu, 31 Oct 2013 00:00:00 +0000The authors present a new, observationally based estimate of the atmospheric energy budget for the Antarctic polar cap (the region poleward of 70°S). This energy budget is constructed using state-of-the-art reanalysis products from ECMWF [the ECMWF Interim Re-Analysis (ERA-Interim)] and Clouds and the Earth's Radiant Energy System (CERES) top-of-atmosphere (TOA) radiative fluxes for the period 2001–10. The climatological mean Antarctic energy budget is characterized by an approximate balance between the TOA net outgoing radiation and the horizontal convergence of atmospheric energy transport, with the net surface energy flux and atmospheric energy storage generally being small in comparison. Variability in the energy budget on intraseasonal-to-interannual time scales bears a strong signature of the southern annular mode (SAM), with El Niño–Southern Oscillation (ENSO) having a smaller impact. The energy budget framework is shown to be a useful alternative to the SAM for interpreting surface climate variability in the Antarctic region.Atmospheric sciences, Meteorology, Applied mathematicsmp2609, kls2177, lmp3Ocean and Climate Physics, Applied Physics and Applied Mathematics, Lamont-Doherty Earth Observatory, Earth and Environmental SciencesArticlesResource Cost Aware Scheduling Problems
http://academiccommons.columbia.edu/catalog/ac:166566
Carrasco, Rodrigohttp://hdl.handle.net/10022/AC:P:21999Thu, 17 Oct 2013 00:00:00 +0000Managing the consumption of non-renewable and/or limited resources has become an important issue in many different settings. In this dissertation we explore the topic of resource cost aware scheduling. Unlike the purely scheduling problems, in the resource cost aware setting we are not only interested in a scheduling performance metric, but also the cost of the resources consumed to achieve a certain performance level. There are several ways in which the cost of non-renewal resources can be added into a scheduling problem. Throughout this dissertation we will focus in the case where the resource consumption cost is added, as part of the objective, to a scheduling performance metric such as weighted completion time and weighted tardiness among others. In our work we make several contributions to the problem of scheduling with non-renewable resources. For the specific setting in which only energy consumption is the important resource, our contributions are the following. We introduce a model that extends the previous energy cost models by allowing more general cost functions that can be job-dependent. We further generalize the problem by allowing arbitrary precedence constraints and release dates. We give approximation algorithms for minimizing an objective that is a combination of a scheduling metric, namely total weighted completion time and total weighted tardiness, and the total energy consumption cost. Our approximation algorithm is based on an interval-and-speed-indexed IP formulation. We solve the linear relaxation of this IP and we use this solution to compute a schedule. We show that these algorithms have small constant approximation ratios. Through experimental analysis we show that the empirical approximation ratios are much better than the theoretical ones and that in fact the solutions are close to optimal. We also show empirically that the algorithm can be used in additional settings not covered by the theoretical results, such as using flow time or an online setting, with good approximation and competitiveness ratios.Industrial engineering, Applied mathematicsIndustrial Engineering and Operations ResearchDissertations