| In this study, theorem-based, generally applicable stochastic parameterisation schemes are developed and applied to a quasi-geostrophic model of extratropical atmospheric low-frequency variability (LFV). Hasselmann's method is developed from limiting theorems for slow-fast systems of ordinary differential equations (ODEs) and applied to this high-dimensional model of intermediate complexity comprised of partial differential equations with complicated boundary conditions. Seamless, efficient algorithms for integrating the parameterised models are developed, which require only minimal changes to the full model algorithm. These algorithms may be readily adapted to a range of climate models of greater complexity in parameterising the effects of fast, sub-grid scale processes on the resolved scales. For comparison, the Majda-Timofeyev-Vanden-Eijnden (MTV) parameterisation method is applied to this model.;The low-dimensional Hasselmann stochastic models perform well in simulating the statistics of the planetary-scale modes. In particular, a model with only one resolved (planetary-scale) mode captures the multiple regime behaviour of the full model. Although a fast-evolving synoptic-scale mode is of primary importance to the multiple regime behaviour, deterministic averaged forcing and not multiplicative noise is responsible for the regime behaviour in this model. The MTV models generate non-Gaussian statistics, but generally do not perform as well in capturing the climate statistics.;The seamless algorithms are first adapted to probe the multiple regime behaviour that characterises the full model LFV. In contrast to the conclusions of a previous study, it is found that the multiple regime behaviour is not the result of a nonlinear interaction between the leading two planetary-scale modes, but rather is the result of interactions among these two modes and the leading synoptic-scale mode. |
