# Closure and order-reduction of turbulent dynamical systems

Turbulent dynamical systems are characterized by both a large dimensional phase space and a large dimension of persistent or intermittent instabilities (i.e., a large number of positive Lyapunov exponents on the attractor). They are ubiquitous in many complex systems with fluid flow such as, engineering turbulence at high Reynolds numbers, confined plasmas, as well as atmospheric and oceanic turbulence (movie below).

The scope of this work is i) to design effective turbulent closure schemes that respect the synergistic activity between unstable linear dynamics, nonlinear energy transfers, and stable dynamics, which is apparent in turbulent systems, and ii) to develop order-reduction techniques that take into account the nonlinear energy transfers associated with the omitted modes.

In these systems the energy of the unstable modes is continuously balanced by nonlinear energy transfer mechanisms that absorb energy from these modes and pass it to the stable ones, resulting in this way, a statistical steady state with broad energy spectrum. From a statistical point view these nonlinear energy transfers are directly connected with the existence of non-Gaussian statistics. Thus, any attempt to design uncertainty quantification algorithms for these systems without taking explicitly into account these nonlinear and non-Gaussian features will result in numerically unstable schemes or schemes that severely underestimate the system energy (to the point where there are no unstable directions in the dynamics).

**Statically accurate Modified Quasilinear Gaussian (MQG) closure for turbulent dynamical systems**

We develop a second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. The resulted UQ scheme provide very accurate estimates of second order statistics even far away from the tuned energy level. Below we observe a comparison of the MQG scheme with direct Monte-Carlo simulations for three different energetic regimes of the Lorenz-96 system. More details can be found here.

**Blending MQG closure and non-Gaussian reduced subspace methods for turbulent systems**

Even though MQG provides with accurate second order statistics even in extreme excitation scenarios, for many applications it is important to know higher order statistical (i.e. expensive) information but only for specific subspaces. In this work we develop a framework that brings together i) inexpensive, second-order modeling for a wide part of the spectrum with ii) higher-order stochastic modeling for a small set of important (possibly adapting, i.e. DO) modes. The coupling of the two approaches is based on arguments involving the nonlinear energy transfers. In the figures below we present the application of this blended approach to the prototype turbulent system Lorenz-96. More details for this work can be found here.

**Radical order-reduction of turbulent systems using reduced-order MQG (ROMQG)**

In both MQG and blended MQG-DO approaches the goal is to model the nonlinear energy fluxes without solving (in MQG) or fully-solving (in MQG-DO) the non-Gaussian character of the statistics (closure problem). Here we go a step further by using these models for the nonlinear energy fluxes to develop radical order reduction methods for turbulent systems, which, despite the very small number of modes considered, are energetically accurate. This is because, in contrast with other approaches, the *emphasis is now given on modeling the energy fluxes to the reduced set of modes as accurate as possible*. [Read more...]

We illustrate the ROMQG algorithm in a two-layer baroclinic turbulence flow with over 125,000 degrees of freedom (see movie on the top of the page as well as the figure below). The inexpensive ROMQG algorithm with 252 modes (0.2% of the total modes) is able to capture the nonlinear response of the energy, the heat flux, and even the one-dimensional, energy and heat flux spectrum at each wavenumber.

**Publications**

`T. Sapsis, A. Majda, `**Statistically Accurate Low Order Models for Uncertainty Quantification in Turbulent Dynamical Systems,** *Proceedings of the National Academy of Sciences*, **110** (2013) 13705-13710. [pdf]

`T. Sapsis, A. Majda, `**Blending modified Gaussian closure and non-Gaussian reduced subspace methods for turbulent dynamical systems**, *Journal of Nonlinear Science,* DOI 10.1007/s00332-013-9178-1 (2013). [pdf]

`T. Sapsis, A. Majda, `**Blended reduced subspace algorithms for uncertainty quantification of quadratic systems with a stable mean state**, *Physica D*, **258** (2013) 61. [pdf]

`T. Sapsis, A. Majda, `**A statistically accurate modified quasilinar Gaussian closure for uncertainty quantification in turbulent dynamical systems**, *Physica D*, **252** (2013) 34. [pdf]