We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.

Coherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number $\mathrm{Ra}={10}^6$ and at a Prandtl number $\mathrm{Pr}=10$. Three different Lagrangian analysis techniques based on graph Laplacians (distance spectral trajectory clustering, time-averaged diffusion maps, and finite-element based dynamic Laplacian discretization) are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport.

Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.

We propose to compute approximations to invariant sets in dynamical systems by minimizing an appropriate distance between a suitably selected finite set of points and its image under the dynamics. We demonstrate, through computational experiments, that this approach can successfully converge to approximations of (maximal) invariant sets of arbitrary topology, dimension, and stability, such as, e.g., saddle type invariant sets with complicated dynamics. We further propose to extend this approach by adding a Lennard-Jones type potential term to the objective function, which yields more evenly distributed approximating finite point sets, and illustrate the procedure through corresponding numerical experiments.

We generalize leading-order asymptotics of a form of the heat content of a submanifold (van den Berg & Gilkey 2015) to the setting of time-dependent diffusion processes in the limit of vanishing diffusivity. Such diffusion processes arise naturally when advection–diffusion processes are viewed in Lagrangian coordinates. We prove that as diffusivity ɛ goes to zero, the diffusive transport out of a material set S under the time-dependent, mass-preserving advection–diffusion equation with initial condition given by the characteristic function 1 S , is ε / π d A ¯ ( ∂ S ) + o ( ε ) . The surface measure d A ¯ is that of the so-called geometry of mixing, as introduced in (Karrasch & Keller 2020). We apply our result to the characterisation of coherent structures in time-dependent dynamical systems.