El Niño-Southern Oscillation (ENSO) has two major facets, namely the Eastern Pacific (EP) and the Central Pacific (CP) events, with irregular and quasi-periodic anomalies in wind currents and sea surface temperatures (SST) making it the most important climate phenomenon in the region. It also exhibits diverse characteristics in spatial pattern, peak intensity, and temporal evolution during its mature warming phase (El Niño) and mature cooling phase (La Niña), known as the ENSO diversity or complexity. Traditional methods for studying the sensitivity and response of ENSO to initial value and model parameter perturbations, are primarily based on trajectory-wise comparison. However, the intrinsic chaotic features and the model error impose significant challenges for accurately computing the statistical response in this manner. In this talk, we present a new approach to calculating the statistical response of ENSO diversity using information theory, by quantifying the intrinsic predictability and its response through a multiscale three-region stochastic model as a surrogate. It computes the response of the statistics, such as the mean and variance, to initial value or model parameter perturbations. We provide the most dangerous direction under initial and parameter perturbations for different ENSO events over the past 36 years (1982-2017). We also show that the uncertainty described by the variance and higher-order moments can have a significant response on certain perturbations, despite the insignificant change in the mean, which is a fundamental mechanism of the increment of extreme El Niño events and multi-year El Niño and La Niña, and that under a univariate SSTa regime for the probability densities, a Gaussian approximation captures most of the intricacies of intrinsic predictability and statistical response. This way of quantifying statistical response is more robust and physically meaningful, since it provides ways to inspect the response of ENSO diversity under the climate change scenario, increase or decrease of the Madden-Julian oscillation (MJO) or tropical cyclones, through model parameter perturbations, or to probe into the principal directions relating to forecast and prevention of extreme events, as well as impact on other climate variabilities, through initial value perturbations.

Souvla, which can be summed up as a chunkier version of the Greek souvlaki, is widely considered to be Cyprus' national dish. Large chunks of lamb or pork meat are pierced with a long metallic skewer, and cooked above a rectangular grill, known as foukou, by rotating the skewer using a motor. In this work we model the cooking process by initially solving the heat equation with rotating Dirichlet boundary conditions, used to simulate heat transfer through pure induction. This includes the computation of a zeroth-order accurate singular perturbation asymptotic expansion of the solution, as the angular velocity of the skewer tends to infinity, as well as an analytic expression for the theoretical cooking time (time it takes for the meat to reach the desired cooking temperature), which we validate via numerical simulations using the Python spectral solver library, Dedalus. We then expand on these findings, by extending our model to a Taylor-Couette flow heat transfer model, where the grill now surrounds the meat on one of its "sides", which leads to a heat equation with rotating Robin boundary conditions, simulating heat transfer at the meat boundary through convection (via the Boussinesq approximation of natural convection) and radiation (from the grill), after a Stefan-Boltzmann linearisation. In this elaborate setting, we again produce in the same manner a zeroth-order accurate singular perturbation asymptotic expansion of the solution and a theoretical cook-through time, as the angular velocity grows unboundedly, using a first Fourier mode approximation attributed to the sparse spectrum. This cook-through time is again validated numerically in Dedalus, by using a mixture of the Diffuse-domain method (DDM) and the Volume-penalty method (VPM) to solve the double domain heat transfer Taylor-Couette flow setting, which is driven by a combination of multiple time scales inside and outside the meat, fluid and temperature damping scales, and boundary layers developing at the meat surface.

State estimation is a critical task in practice as it is the prerequisite for effective parameter estimation (PE), skillful prediction, optimal control, and the development of robust and complete datasets. Of essence is the probabilistic state estimation of the unobserved variables in high-dimensional complex nonlinear turbulent dynamical systems with intermittent instability, given a partially observed time series. Data assimilation (DA) through Bayesian inference combines the partial observations and the model-induced likelihood, potentially plagued by a small error in model structure or initial condition, to form the posterior distribution for optimal state estimation. In this work, an efficient algorithm with closed-form explicit expressions is developed for the optimal (in the mean-squared sense) online smoother and sampling of a rich class of nonlinear turbulent dynamical systems widely appearing in modeling natural phenomena like physics-constrained stochastic models (noisy Lorenz models, low-order models of Charney-DeVore flows, paradigm models for topographic mean flow interaction), stochastically coupled reaction-diffusion models in neuroscience and ecology (stochastically coupled FitzHugh-Nagumo models, stochastically coupled SIR epidemic models), multiscale models for geophysical flows (noisy Boussinesq equations, stochastically forced rotating shallow water equation), and to develop realistic systems for the Madden-Julian oscillation and Arctic sea ice. The conditional Gaussian nonlinear system framework is highly flexible since it encompasses the linear Kalman filter and the Rauch-Tung-Striebel smoother and facilitates studying extreme events, stochastic parameterisation, DA, and online PE. The method particularly handles systems with hidden intermittency and extreme events along with the associated highly non-Gaussian features, such as heavy tails in the probability density functions, by using a computationally effective and systematic method to determine the adaptive lag for the online procedure. A rigorous analysis of the performance of the framework will be illustrated through numerical simulations of high-dimensional Lagrangian DA and online model identification of highly intermittent time series via an expectation-maximization procedure.