Computation of rare transitions in the barotropic quasi-geostrophic equations

We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherWe adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems

Six-wave systems in one-dimensional wave turbulence

We investigate one-dimensional (1D) wave turbulence (WT) systems that are characterised by six-wave interactions. We begin by presenting a brief introduction to WT theory - the study of the non-equilibrium statistical mechanics of nonlinear random waves, by giving a short historical review followed by a discussion on the physical applications. We implement the WT description to a general six-wave Hamiltonian system that contains two invariants, namely, energy and wave action. This enables the subsequent derivations for the evolutions equations of the one-mode amplitude probability density function (PDF) and kinetic equation (KE). Analysis of the stationary solutions of these equations are made with additional checks on their underlying assumptions for validity. Moreover, we derive a differential approximation model (DAM) to the KE for super-local wave interactions and investigate the possible occurrence of a fluctuation relation. We then consider these results in the context of two physical systems - Kelvin waves in quantum turbulence (QT) and optical wave turbulence (OWT). We discuss the role of Kelvin waves in decaying QT, and show that they can be described by six-wave interactions. We explicitly compute the interaction coefficients for the Biot-Savart equation (BSE) Hamiltonian and represent the Kelvin wave dynamics in the form of a KE. The resulting non-equilibrium Kolmogorov-Zakharov (KZ) solutions to the KE are shown to be non-local, thus a new non-local theory for Kelvin wave interactions is discussed. A local equation for the dynamics of Kelvin waves, the local nonlinear equation (LNE), is derived from the BSE in the asymptotic limit of one long Kelvin wave. Numerical computation of the LNE leads to an agreement with the nonlocal Kelvin wave theory. Finally, we consider 1D OWT. We present the first experimental implementation of OWT and provide a comparable decaying numerical simulation for verification. We show that 1D OWT is described by a six-wave process and that the inverse cascade state leads to the development of coherent solitons at large scales. Further investigation is conducted into the behaviour of solitons and their impact to the WT description. Analysis of the fluxes and intensity PDFs lead to the development of a wave turbulence life cycle (WTLC), explaining the coexistence between coherent solitons and incoherent waves. Additional numerical simulations are performed in non-equilibrium stationary regimes to determine if a pure KZ state can be realised

Coarse-grained pressure dynamics in superfluid turbulence

Quantum mechanics places significant restrictions on the hydrodynamics of superfluid flows. Despite this it has been observed that turbulence in superfluids can, in a statistical sense, share many of the properties of its classical brethren; coherent bundles of superfluid vortices are often invoked as an important feature leading to this quasiclassical behavior. A recent experimental study [E. Rusaouen, B. Rousset, and P.-E. Roche, Europhys. Lett. 118, 14005, (2017)10.1209/0295-5075/118/14005] inferred the presence of these bundles through intermittency in the pressure field; however, direct visualization of the quantized vortices to corroborate this finding was not possible. In this work, we performed detailed numerical simulations of superfluid turbulence at the level of individual quantized vortices through the vortex filament model. Through course graining of the turbulent fields, we find compelling evidence supporting these conclusions at low temperature. Moreover, elementary simulations of an isolated bundle show that the number of vortices inside a bundle can be directly inferred from the magnitude of the pressure dip, with good theoretical agreement derived from the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations. Full simulations of superfluid turbulence show strong spatial correlations between course-grained vorticity and low-pressure regions, with intermittent vortex bundles appearing as deviations from the underlying Maxwellian (vorticity) and Gaussian (pressure) distributions. Finally, simulations of a decaying random tangle in an ultraquantum regime show a unique fingerprint in the evolution of the pressure distribution, which we argue can be fully understood using the HVBK framework

A perspective on trends in Australian government spending

This paper provides a summary of trends in government spending, revealing strong growth in government spending and the size of government, particularly over the past four years. It also discusses the distribution and sustainability of spending and notes the importance of high quality spending and flexibility in resource allocation in responding to future pressures

Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors

Jets or vortices - what flows are generated by an inverse turbulent cascade?

An inverse cascade{energy transfer to progressively larger scales{is a salient feature of two-dimensional turbulence. If the cascade reaches the system scale, it creates a coherent ow expected to have the largest available scale and conform with the symmetries of the domain. In a doubly periodic rectangle, the mean ow with zero total momentum was therefore believed to be unidirec- tional, with two jets along the short side; while for an aspect ratio close to unity, a vortex dipole was expected. Using direct numerical simulations, we show that in fact neither the box symmetry is respected nor the largest scale is realized: the ow is never purely unidirectional since the inverse cascade produces coherent vortices, whose number and relative motion are determined by the aspect ratio. This spontaneous symmetry breaking is closely related to the hierarchy of averaging times. Long-time averaging restores translational invariance due to vortex wandering along one direction, and gives jets whose profile, however, can be deduced neither from the largest-available-scale argument, nor from the often employed maximum-entropy principle or quasilinear approximation

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum

Does self-modulated learning vs. algorithm-regulated learning of dermatology morphology affect learning efficiency of medical students?

Background: Deliberate practice is an important method of skill acquisition and is under-utilized in dermatology training. We delivered a dermatologic morphology training module with immediate feedback for first year medical students. Our goal was to determine whether there are differences in accuracy and learning efficiency between self-regulated and algorithm-regulated groups. Methods: First year medical students at the University of Calgary completed a dermatologic morphology module. We randomly assigned them to either a self-regulated arm (students removed cases from the practice pool at their discretion) or an algorithm-regulated arm (an algorithm determined when a case would be removed). We then administered a pre-survey, pre-test, post-test, and post-survey. Data collected included mean diagnostic accuracy of the practice sessions and tests, and the time spent practicing. The surveys assessed demographic data and student satisfaction. Results: Students in the algorithm-regulated arm completed more cases than the self-regulated arm (52.9 vs. 29.3, p<0.001) and spent twice as much time completing the module than the self-regulated participants (34.3 vs. 17.0 min., p<0.001). Mean scores were equivalent between the algorithm- and self-regulated groups for the pre-test (63% vs. 66%, n = 54) and post-test (90% vs. 86%, n = 10), respectively. Both arms demonstrated statistically significant improvement in the post-test. Conclusion: Both the self-regulated and algorithm-regulated arms improved at post-test. Students spent significantly less time practicing in the self-directed arm, suggesting it was more efficient

Six-wave systems in one-dimensional wave turbulence

We investigate one-dimensional (1D) wave turbulence (WT) systems that are characterised by six-wave interactions. We begin by presenting a brief introduction to WT theory - the study of the non-equilibrium statistical mechanics of nonlinear random waves, by giving a short historical review followed by a discussion on the physical applications. We implement the WT description to a general six-wave Hamiltonian system that contains two invariants, namely, energy and wave action. This enables the subsequent derivations for the evolutions equations of the one-mode amplitude probability density function (PDF) and kinetic equation (KE). Analysis of the stationary solutions of these equations are made with additional checks on their underlying assumptions for validity. Moreover, we derive a differential approximation model (DAM) to the KE for super-local wave interactions and investigate the possible occurrence of a fluctuation relation. We then consider these results in the context of two physical systems - Kelvin waves in quantum turbulence (QT) and optical wave turbulence (OWT). We discuss the role of Kelvin waves in decaying QT, and show that they can be described by six-wave interactions. We explicitly compute the interaction coefficients for the Biot-Savart equation (BSE) Hamiltonian and represent the Kelvin wave dynamics in the form of a KE. The resulting non-equilibrium Kolmogorov-Zakharov (KZ) solutions to the KE are shown to be non-local, thus a new non-local theory for Kelvin wave interactions is discussed. A local equation for the dynamics of Kelvin waves, the local nonlinear equation (LNE), is derived from the BSE in the asymptotic limit of one long Kelvin wave. Numerical computation of the LNE leads to an agreement with the nonlocal Kelvin wave theory. Finally, we consider 1D OWT. We present the first experimental implementation of OWT and provide a comparable decaying numerical simulation for verification. We show that 1D OWT is described by a six-wave process and that the inverse cascade state leads to the development of coherent solitons at large scales. Further investigation is conducted into the behaviour of solitons and their impact to the WT description. Analysis of the fluxes and intensity PDFs lead to the development of a wave turbulence life cycle (WTLC), explaining the coexistence between coherent solitons and incoherent waves. Additional numerical simulations are performed in non-equilibrium stationary regimes to determine if a pure KZ state can be realised.EThOS - Electronic Theses Online ServiceGBUnited Kingdo