An applied mathematics perspective on stochastic modelling for climate

Abstract

Systematic strategies from applied mathematics for stochastic modelling in climate are reviewed here. One of the topics discussed is the stochastic modelling of mid-latitude low-frequency variability through a few teleconnection patterns, including the central role and physical mechanisms responsible for multiplicative noise. A new lowdimensional stochastic model is developed here, which mimics key features of atmospheric general circulation models, to test the fidelity of stochastic mode reduction procedures. The second topic discussed here is the systematic design of stochastic lattice models to capture irregular and highly intermittent features that are not resolved by a deterministic parametrization. A recent applied mathematics design principle for stochastic column modelling with intermittency is illustrated in an idealized setting for deep tropical convection; the practical effect of this stochastic model in both slowing down convectively coupled waves and increasing their fluctuations is presented here.

Stochastic models for convective momentum transport

“…The stochastic model for CMT developed here involves a 3-state continuous-time Markov jump process (25) for the small-scale dynamics at each large-scale spatiotemporal location (x, t) with transition rates depending in a suitable fashion on the local values of the large-scale variables at (x, t) (20)(21)(22)(23)(24). Depending on the current small-scale state, the strength and nature of the CMT in Eq.…”

“…Theory and applications for such types of coarse-grained stochastic models have been developed recently (20)(21)(22)(23)(24). In fact, it is increasingly apparent that suitable stochastic parameterization is an important strategy to improve the fidelity of unresolved features in contemporary weather and climate models for a variety of physical processes (24). Here, we develop and test a simple stochastic model for CMT that accounts for the novel physical features of energy transfer in this context.…”

“…These are exactly the types of circumstances where suitable coarse-grained stochastic models are able to capture the intermittent impact of smaller-scale events on the larger scales. Theory and applications for such types of coarse-grained stochastic models have been developed recently (20)(21)(22)(23)(24). In fact, it is increasingly apparent that suitable stochastic parameterization is an important strategy to improve the fidelity of unresolved features in contemporary weather and climate models for a variety of physical processes (24).…”

Seasonal and decadal forecasts of Atlantic Sea surface temperatures using a linear inverse model

“…LIM is a reduced system of equations of a complete stochastic-dynamic model for the climate (Majda et al 1999(Majda et al , 2008. LIM has been used for forecast skill evaluation of global, tropical and extratropical weather and climate (Pegion and Sardeshmukh 2011;Vimont 2012;Zanna 2012;Newman 2013).…”

Maximum likelihood estimation for small noise multiscale diffusions

“…Modeling this type of data via diffusion type models may be well-suited in many cases. Thus, multiscale diffusion models have been used to describe the behavior of physical phenomena in scientific areas such as chemistry and biology [6,15,23,26], ocean-atmosphere sciences [20], finance and econometrics [1,14]. In many of these problems, the noise is taken to be small because one may, for example, be interested in modeling (a): rare 1 transition events between equilibrium states of a rough energy landscape [8,15,26], or (b): short time maturity asymptotics for fast mean reverting stochastic volatility models [10,11].…”