Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyze deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.
Abstract Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we establish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for \SPDEs\ are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness results and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived.
The comparison performed in Berry et al. [Phys. Rev. E 91, 032915 (2015)] between the skill in predicting the El Niño-Southern Oscillation climate phenomenon by the prediction method of Berry et al. and the “past-noise” forecasting method of Chekroun et al. [Proc. Natl. Acad. Sci. USA 108, 11766 (2011)] is flawed. Three specific misunderstandings in Berry et al. are pointed out and corrected.
A suite of empirical model experiments under the empirical model reduction framework are conducted to advance the understanding of
ENSO diversity, nonlinearity, seasonality, and the memory effect in the simulation and prediction of tropical Pacific sea surface temperature (SST) anomalies. The model training and evaluation are carried out using 4000-yr preindustrial control simulation data from the coupled model GFDL CM2.1. The results show that multivariate models with tropical Pacific subsurface information and multilevel models with SST history information both improve the prediction skill dramatically. These two types of models represent the ENSO memory effect based on either the recharge oscillator or the time-delayed oscillator viewpoint. Multilevel SST models are a bit more efficient, requiring fewer model coefficients. Nonlinearity is found necessary to reproduce the ENSO diversity feature for extreme events. The nonlinear models reconstruct the skewed probability density function of SST anomalies and improve the prediction of the skewed amplitude, though the role of nonlinearity may be slightly overestimated given the strong nonlinear ENSO in GFDL CM2.1. The models with periodic terms reproduce the SST seasonal phase locking but do not improve the prediction appreciably. The models with multiple ingredients capture several ENSO characteristics simultaneously and exhibit overall better prediction skill for more diverse target patterns. In particular, they alleviate the spring/autumn prediction barrier and reduce the tendency for predicted values to lag the target month value.