Nonlinear Stochastic Low-Order Models of Atmospheric Low-Frequency Variability

Nonlinear stochastic low-order models are derived that are capable of simulating realistic atmospheric low-frequency variability (Kwasniok 2005b). The reduced models deal explicitly only with a very limited number of essential low-frequency patterns; the influence of unresolved fast-evolving modes onto the resolved slowly- evolving large-scale modes is modeled by stochastic terms. A strategy of keeping the number of fitting parameters in the dynamical equations to a minimum is applied. The study focuses on two issues: (i) finding appropriate basis functions for spanning the resolved dynamics and a comparison between different choices of basis functions; (ii) the necessary characteristics of the noise processes, that is, white vs. colored noise, Gaussian vs. non-Gaussian noise, and additive vs. multiplicative noise. A quasi- geostrophic three-level spectral model, truncated to T30, with realistic variability is used as dynamical framework.

Firstly, a barely deterministic reduced model is obtained by projecting the equations of motion onto a truncated basis spanned by empirically determined patterns. The total energy metric is used in the projection; the nonlinear terms of the low-order model then conserve total energy. No empirical terms are fitted in the dynamical equations of the low-order model in this step in order to properly preserve the physics of the system. Two choices of empirical patterns are considered: the conventional empirical orthogonal functions (EOFs) and the dynamically motivated principal interaction patterns (PIPs) (Crommelin and Majda 2004; Kwasniok 2004, 2005a). The PIP approach is tailored in a way to keep it computationally feasible for the high-dimensional dynamical framework under consideration here (Kwasniok 2005a). Then the bare truncation is augmented by additive Gaussian stochastic terms whose properties are obtained empirically from time series of the tendency errors of the deterministic low-order model. This yields a Langevin equation for the resolved modes. In a first step, white noise is assumed; its covariance structure is determined from the data. The coefficient of horizontal diffusion is retuned empirically as a function of the truncation level in a way that the stochastic reduced model in a long-term integration exhibits the same amount of variance in the resolved modes as is present in the full spectral model. In a further step, the remaining autocorrelation in the tendency errors is taken into account by approximating the tendency errors by an autoregressive process of first order, thus introducing one further parameter for each resolved mode.

Long-term integrations of the stochastic models are performed at various truncation levels and their long-term behavior is compared with that of the full model. Monitored quantities include the mean state, the variance pattern, momentum fluxes, probability distributions and autocorrelation functions. Stochastic low-order models (already with white noise) greatly outperform purely deterministic low-order models. A Langevin equation with only 10 retained low-frequency patterns is able to self-consistently simulate the dynamics of these patterns whereas a deterministic model with this few modes is completely off. The extension from white to red noise offers a considerable further improvement. With both kinds of noise processes, models based on PIPs clearly outperform models based on EOFs.