Terence J. O'Kane

Stochastic climate theory and modeling

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models.

The QDIA and regularized QDIA closures for inhomogeneous turbulence over topography

The dynamics and spectra of the quasi-diagonal direct interaction approximation (QDIA) closure for inhomogeneous two-dimensional turbulence over mean (single realization) topography are compared with results from direct numerical simulations (DNS). A more efficient version of the closure, termed the cumulant update QDIA (CUQDIA), has also been formulated and tested. Studies are performed for a range of resolutions, for large scale Reynolds numbers between very low (

Statistical dynamical subgrid-scale parameterizations for geophysical flows

R_{L} ) and moderate (

Sources of heterogeneous variability and trends in Antarctic sea-ice

R_{L} \approxeq 300

Stochastic subgrid parameterizations for atmospheric and oceanic flows

) and for wide ranges of topographic spectra and initial mean field and transient spectra. The QDIA-type closures are shown to be computationally tractable for general inhomogeneous flows, particularly in cumulant update form, and to perform extremely well when the turbulence is weak. At low (

ENSO regimes and the late 1970's climate shift

R_{L} \approxeq 60

Subgrid modelling for geophysical flows

) to moderate (

Application of statistical dynamical turbulence closures to data assimilation

R_{L} \approxeq 300

Stochastic Subgrid Modelling for Geophysical and Three-Dimensional Turbulence

) Reynolds numbers the presence of significant amplitude small-scale mean fields and topography reduces the under-estimation of small-scale transient kinetic energy that is characteristic of the Eulerian direct interaction approximation (DIA). A regularized version of the CUQDIA closure (RCUQDIA) in which interactions are localized in wavenumber space, depending on specified cut-off ratios, has also been tested at moderate Reynolds number for cases when the small-scale mean fields and topography are weak. Excellent agreement has been found between the RCUQDIA closure and DNS results for turbulent flows with properties broadly similar to atmospheric spectra.

TURBULENCE CLOSURES AND SUBGRID-SCALE PARAMETERIZATIONS

Simulations of both atmospheric and oceanic circulations at given finite resolutions are strongly dependent on the form and strengths of the dynamical subgrid-scale parameterizations (SSPs) and in particular are sensitive to subgrid-scale transient eddies interacting with the retained scale topography and the mean flow. In this paper, we present numerical results for SSPs of the eddy–topographic force, stochastic backscatter, eddy viscosity and eddy–mean field interaction using an inhomogeneous statistical turbulence model based on a quasi-diagonal direct interaction approximation (QDIA). Although the theoretical description on which our model is based is for general barotropic flows, we specifically focus on global atmospheric flows where large-scale Rossby waves are present. We compare and contrast the closure-based results with an important earlier heuristic SSP of the eddy–topographic force, based on maximum entropy or statistical canonical equilibrium arguments, developed specifically for general ocean circulation models (Holloway 1992 J. Phys. Oceanogr. 22 1033–46). Our results demonstrate that where strong zonal flows and Rossby waves are present, such as in the atmosphere, maximum entropy arguments are insufficient to accurately parameterize the subgrid contributions due to eddy–eddy, eddy–topographic and eddy–mean field interactions. We contrast our atmospheric results with findings for the oceans. Our study identifies subgrid-scale interactions that are currently not parameterized in numerical atmospheric climate models, which may lead to systematic defects in the simulated circulations.