We describe a new method for computing coherent Lagrangian vortices in two-dimensional flows according to any of the following approaches: blackhole vortices [23], objective Eulerian Coherent Structures (OECSs) [37], material barriers to diffusive transport [24,25], and constrained diffusion barriers [25]. The method builds on ideas developed previously in [29], but our implementation alleviates a number of shortcomings and allows for the fully automated detection of such vortices on unprecedentedly challenging real-world flow problems, for which specific human interference is absolutely infeasible. Challenges include very large domains and/or parameter spaces. We demonstrate the efficacy of our method in dealing with such challenges on two test cases: first, a parameter study of a turbulent flow, and second, computing material barriers to diffusive transport in the global ocean.

In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like S exp(t)(f 1 S) dV = S f dV − t π ∂ S f d A + o(√ t), t → 0 + , and explicit expressions for similar expansions involving other powers of √ t. By the same method, we also obtain short-time asymptotics of S exp(t m m)(f 1 S) dV , m ∈ N, and more generally for one-parameter families of operators t → k(√ −t) defined by an even Schwartz function k.

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourthorder accurate Runge-Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge-Kutta pair of Bogacki & Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy. *

The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in [14]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

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.