Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM

Transport in time-dependent dynamical systems: Finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite time duration. We are particularly interested in those regions that remain coherent and relatively non-dispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detects maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data

Perturbation dynamics of a planktonic ecosystem

Planktonic ecosystems provide a key mechanism for the transfer of carbon from the atmosphere to the deep ocean via the so-called biological pump. Mathematical models of these ecosystems have been used to predict CO2 uptake in surface waters at particular locations, and more recently have been embedded in global climate models. While the equilibrium properties of these models are well studied, less attention has been paid to their response to external perturbations, despite the fact that as a result of the variability of environmental forcing such ecosystems are rarely, if ever, in equilibrium. In this study, linear theory is used to determine the structure of perturbations to state variables of an ecosystem model describing summertime conditions at Ocean Station P (50°N 145°W) that maximize either instantaneous or integrated export flux. As a result of the presence of both direct and indirect pathways to export in this model, these perturbations involve the dynamics of the entire ecosystem. For all optimal perturbations considered, it is found that the flux to higher trophic levels is the primary contributor to export flux, followed by sinking detritus. In contrast, the contribution of aggregation is negligible. In addition, small phytoplankton contribute significantly (comparable to large phytoplankton) to the export flux through indirect pathways, primarily through the microzooplankton, even following a bloom in only large phytoplankton. While the details of these results may be specific to the particular model under consideration, the optimal perturbation framework is general and can be used to probe the dynamics of any mechanistic ecosystem model

Teleconnected warm and cold extremes of North American wintertime temperatures

Current models for spatial extremes are concerned with the joint upper (or lower) tail of the distribution at two or more locations. Such models cannot account for teleconnection patterns of two-meter surface air temperature ($T_{2m}$) in North America, where very low temperatures in the contiguous Unites States (CONUS) may coincide with very high temperatures in Alaska in the wintertime. This dependence between warm and cold extremes motivates the need for a model with opposite-tail dependence in spatial extremes. This work develops a statistical modeling framework which has flexible behavior in all four pairings of high and low extremes at pairs of locations. In particular, we use a mixture of rotations of common Archimedean copulas to capture various combinations of four-corner tail dependence. We study teleconnected $T_{2m}$ extremes using ERA5 reanalysis of daily average two-meter temperature during the boreal winter. The estimated mixture model quantifies the strength of opposite-tail dependence between warm temperatures in Alaska and cold temperatures in the midlatitudes of North America, as well as the reverse pattern. These dependence patterns are shown to correspond to blocked and zonal patterns of mid-tropospheric flow. This analysis extends the classical notion of correlation-based teleconnections to considering dependence in higher quantiles

Robustness of the stochastic parameterization of sub-grid scale wind variability in sea-surface fluxes

High-resolution numerical models have been used to develop statistical models of the enhancement of sea surface fluxes resulting from spatial variability of sea-surface wind. In particular, studies have shown that the flux enhancement is not a deterministic function of the resolved state. Previous studies focused on single geographical areas or used a single high-resolution numerical model. This study extends the development of such statistical models by considering six different high-resolution models, four different geographical regions, and three different ten-day periods, allowing for a systematic investigation of the robustness of both the deterministic and stochastic parts of the data-driven parameterization. Results indicate that the deterministic part, based on regressing the unresolved normalized flux onto resolved scale normalized flux and precipitation, is broadly robust across different models, regions, and time periods. The statistical features of the stochastic part of the model (spatial and temporal autocorrelation and parameters of a Gaussian process fit to the regression residual) are also found to be robust and not strongly sensitive to the underlying model, modelled geographical region, or time period studied. Best-fit Gaussian process parameters display robust spatial heterogeneity across models, indicating potential for improvements to the statistical model. These results illustrate the potential for the development of a generic, explicitly stochastic parameterization of sea-surface flux enhancements dependent on wind variability

Stochastic Stability of Open-Ocean Deep Convection

Open-ocean deep convection is a highly variable and strongly nonlinear process that plays an essential role in the global ocean circulation. A new view of its stability is presented here, in which variability, as parameterised by stochastic forcing, is central. The use of an idealised deep convection box model allows analytical solutions and straightforward conceptual understand-ing, while retaining the main features of deep convection dynamics. In contrast to the generally abrupt stability changes in deterministic systems, measures of stochastic stability change smoothly in response to varying forcing parameters. These stochastic stability measures depend chie y on the residence times of the system in dierent regions of phase space, which need not contain a stable steady state in the deterministic sense. Deep convection can occur frequently even for parameter ranges in which it is deterministically unstable; this eect is denoted wandering unimodality. The stochastic stability concepts are readily applied to other components of the climate system. The results highlight the need to take climate variability into account when analysing the stability of a climate state.