Stochastic Closure for Large-Scale Turbulent Flows
Presiding: D Livescu, Los Alamos National Laboratory; B Nadiga, Los Alamos National Laboratory
NG43A-01 INVITED 13:30h
Systematic Reduced Stochastic Climate Models of Atmospheric Low-Frequency Variability
This study applies a new systematic, mathematical strategy for stochastic climate modeling for atmospheric low-frequency variability. This strategy is motivated by the fact that atmospheric low-frequency variability can be efficiently described by only a few dominant teleconnection patterns or basis functions. The stochastic modeling strategy is applied to a set of global circulation models with increasing complexity which simulate the observed winter circulation well. In particular, results from a global barotropic model and a global 3 layer quasi-geostrophic model will be presented. The systematic strategy, developed by Majda et al. (1999, 2001, 2002, 2003) and Franzke et al. (2005) consists first of the identification of slowly evolving climate modes and faster evolving non-climate modes by use of an empirical orthogonal function decomposition and by minimal regression fitting of the unresolved modes. The stochastic climate model predicts the evolution of these climate modes only. Since the climate system is governed by nonlinear equations the interactions of the resolved climate modes with the unresolved non-climate modes have to be taken into account. The low-order stochastic climate model predicts the evolution of these climate modes a priori without any regression fitting of the resolved modes. The systematic stochastic mode reduction strategy determines all correction terms and noises with minimal regression fitting of the variances and correlation times of the unresolved modes. These correction terms and noises account for the neglected interactions between the resolved climate modes and the unresolved non-climate modes. No ad hoc damping is necessary as in previous studies. All additional interaction terms are predicted which include constant forcing terms, linear terms, quadratic and cubic nonlinear terms, as well as additive and multiplicative (state dependent) noises. These additional interaction terms describe the interaction of the resolved with the unresolved modes in a rigorous systematic way. The stochastic models reproduce the geographical distributions of the variances and transient eddy forcing well. Also the decay of the autocorrelation functions and the PDFs are captured reasonably well. These results provide evidence of effective stochastic dynamics in climate. Furthermore, the stochastic mode reduction strategy reveals fundamental differences between the barotropic and the baroclinic models. While the reduced stochastic model of the barotropic model is essentially linear with additive noise, the reduced stochastic model of the 3 layer quasi-geostrophic model is dominated by both linear and nonlinear dynamics and by both additive and multiplicative noises. The dynamical implications of these differences as well as different optimal basis function strategies will be discussed.
NG43A-02 14:00h
A Dynamic Closure of Synoptic Eddy and Low-frequency Flow (SELF) Interaction and the Self-organization of Low-frequency Modes
The two-way interaction between synoptic eddy and low-frequency flow (SELF), which we will refer to as the SELF interaction, has been recognized for decades to play an important role in the dynamics of the low-frequency variability of the atmospheric circulation. We propose a new framework for studying the dynamics of the SELF interaction and the low-frequency variability in a stormy background flow. By considering a Gaussian flow as a surrogate for the stormy background flow, we expand the traditional climatological basic flow to a synthetic stochastic basic flow. Its ensemble mean is the observed climatological mean flow while its prescribed variance/covariance fields represent the climatological variance/covariance fields of the observed synoptic eddies. Low-frequency anomalies in the traditional month-to-seasonal mean flow and in the variance/covariance fields of the transient eddy flow are viewed as equivalent to the anomalies in the first and second moments of the quasi-stationary stochastic flow ensemble. The linear dynamics of SELF interaction are described by the coupling among the anomalies in first and second moments. Under the assumption that slow changes in the second moments are in quasi-equilibrium with the anomalies in the first moment, an analytical non-local dynamical closure for SELF interaction is obtained. Using this framework, we show that leading low-frequency modes earn their dominance because they can effective organizing the turbulent synoptic flow such that they get reinforced by positive SELF interaction.
NG43A-03 INVITED 14:30h
A Formula for Estimating the Mean Effects of State Dependent Noise
It is well known that perturbing a dynamical system with stochastic noise can not only alter its variability but also its time mean, especially if the noise is multiplicative, i.e. if its amplitude depends upon the system state. Analytical expressions for this mean effect can be written down for linear systems perturbed by multiplicative white noise. For the practically more relevant case of linear systems perturbed by multiplicative red noise (i.e. noise with a finite correlation time scale), the problem becomes much more difficult. This is because the moment equations in this case are not closed: equations for the lower-order moments involve higher-order moments. Here we introduce a closure approximation that allows the mean effects of the noise in such systems to be estimated with high accuracy. The approximation is accurate for red noises with a wide range of correlation scales, and approaches the correct limits for both very small and very large correlation scales. We illustrate an application to stochastically perturbed planetary-scale Rossby wave dispersion over the globe, and discuss the implications for weather prediction and climate modeling.