# Order reduction and UQ of chaotic flows with low-dimensional attractors - DO equations

It is often the case that infinite-dimensional dynamical systems such as Navier-Stokes equations posses global attractors with low dimensionality, i.e. the set that contains all the possible outcomes of the chaotic system "lives" in a low-dimensional space. For such case it is beneficial to perform order-reduction of the original equation and this is usually done by selecting a set of basis elements or fields (usually based on energetic criteria) and perform a Galerkin projection of the dynamics.

In many situations of practical interest this approach gives powerful results. However, in cases where the problem is strongly transient, even if the number of instabilities is small, it is very hard to choose the appropriate set of modes in order to perform effective order-reduction.

In this work we derive an exact, closed set of evolution equations (the Dynamically Orthogonal (DO) equations - read more...) that allow us to simultaneously evolve i) the basis elements that capture the modes of the flow that have important energy and ii) the statistical structure of the coefficients for these modes (i.e. the shape of the attractor). This new probabilistic framework allows for detailed understanding of the dynamical mechanisms (energy transfers, statistics, instabilities) but also for the efficient uncertainty quantification (UQ) of transient flows as long as low-dimensional attractors characterize these.

*DO stochastic solution for the flow behind a cylinder (Re=100)*

In the movie shown above we present the DO solution for the viscous flow (Re=100) behind a cylinder. The mean flow together with the time-dependent DO modes is visualized in terms of their vorticity field. In addition, the statistics of the stochastic coefficients for the DO modes are shown in terms of their one-dimensional marginals (bottom row) and the 3D joint probability density function for the first three most energetic modes.

In the movies below two more unstable flows are shown: the double gyre flow: an idealized model of the midlatitude wind-driven ocean circulation [Read more...] as well as the lid-driven cavity flow. The number of the DO modes employed in the numerical schemes changes in accordance with the variance of existing modes [Read more for adaptive criteria...]. The adaptive character of the DO modes allows them to focus exactly at the regions where instabilities of the mean flow occur, capturing in this way the important nonlinear energy transfers [Read more...].

We have applied the DO framework to study the stability and energy transfer properties in thermo-fluids and in particular in the Rayleigh Benard convection for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different flow parameters and we reveal the low-dimensionality of the underlying stochastic attractor. [Read more...]

**Attractor local dimensionality and nonlinear energy transfers**

We rigorously examine the geometry of the finite-dimensional attractor associated with fluid flows described by Navierâ€“Stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components (modes) of the flow. The low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms. In particular, we illustrate the connection of the low-dimensionality of the attractor with the circulation of energy: (i) from the mean flow to the unstable modes (due to their linearly unstable character), (ii) from the unstable modes to the stable ones (due to a nonlinear energy transfer mechanism) and (iii) from the stable modes back to the mean (due to the linearly stable character of these modes). [Read more...]

**Publications**

M. Choi, T. Sapsis, G. E. Karniadakis,On the equivalence of dynamically orthogonal and dynamically bi-orthogonal methods: Theory and Numerical simulations,J. Comp. Phys., 333 (2014) 3214-3235. [pdf]

`T. Sapsis, `**Attractor local dimensionality, nonlinear energy transfers, and finite-time instabilities in stochastic dynamical systems with applications to 2D fluid flows**, *Proceedings of the Royal Society A*, **469** (2013) 2153. [pdf]

`T. Sapsis, and H. A. Dijkstra, `**Interaction of noise and nonlinear dynamics in the double-gyre wind-driven ocean circulation, ***Journal of Physical Oceanography*, **43** (2013) 366. [pdf]

`T. Sapsis, M. Ueckermann, P. Lermusiaux, `**Global analysis of Navier-Stokes and Boussinesq stochastic flows using dynamical orthogonality, ***Journal of Fluid Mechanics*, **734** (2013) 83. [pdf]

`M. Choi, T. Sapsis, G. E. Karniadakis, `**Evolution Equations for Time-Evolving Stochastic Modes in Polynomial Chaos - A Convergence Study**, *J. Comp. Phys.*, **245** (2013) 281-301. [pdf]

`M. Ueckermann, P. Lermusiaux, T. Sapsis, `**Numerical Schemes for Dynamically Orthogonal Equations of Stochastic Fluid and Ocean Flows**, *J. Comp. Phys.*, **233** (2013) 272. [pdf]

`T. Sapsis & P. Lermusiaux, `**Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty**, *Physica D*, **241** (2012) 60. [pdf]

`T. Sapsis & P. Lermusiaux, `**Dynamically Orthogonal field equations for continuous stochastic dynamical systems**, *Physica D*, **238** (2009) 2347-2360. [pdf]