Lagrangian methods in 2D annular Rayleigh-Bénard convection

In this project, we approach convective instabilities from the perspective of dynamical systems theory, as we seek to identify structures that organize the global and long-term behavior of a system. Lagrangian Coherent Structures (LCSs) are patterns in fluid flows delineating regions that share a certain notion of material coherence, shape global transport and act as mixing barriers [5]. Thus, characterizing these objectively defined structures allows us to gain new insight into how certain invariant manifolds have a fundamental impact on transport and mixing processes in complex natural environments.

On the other hand, thermal convection turns out to be a fundamental process in geophysical and astrophysical flows by driving large amounts of materials through plumes that allow physical processes to be in constant renewal. Examples are convective cores in massive stars and the interior of planets [1]. It also happens to be a crucial driver of turbulence in even more complicated systems, such as accretion disks [8].

To this end, we present an analysis of coherent structures in convective flows in a particularly unexplored geometry: a 2D annulus under the action of a radial inwardly increasing gravity contribution, g∝1/r (r denotes radius). As disks in astrophysical settings are often modeled as rotating concentric cylinders with small height-to-radius ratio, this simple 2D model allows us to make a fairly global picture of the 3D case with reduced computational cost. Thus, we perform hydrodynamic simulations using spectral tau methods via open-source software Dedalus3 [4]. Equipped with a set of tracer trajectories, we implement different (but complementary) coherent structures approaches, namely objective geometrical techniques such as Finite-Time Lyapunov Exponents (FTLE) and Lagrangian-Averaged Vorticity Deviation (LAVD) [6-7] as well as network-based methods [8].

In this presentation, we will discuss our latest results combining these approaches. We will also make some useful comparisons with [2-3] that complement their Eulerian study in the same geometry.

References

[1] E.H. Anders et al., The Astrophysical Journal, 926, 169 (2022).

[2] A. Bhadra, O. Shiskina, X. Zhu, Journal of Fluid Mechanics, 999, R1 (2024).

[3] A. Bhadra, O. Shiskina, X. Zhu, International Journal of Heat and Mass Transfer, 241, 126703 (2025).

[4] K.J. Burns, G.M. Vasil, J.S. Oishi, D. Lecoanet, B.P. Brown, Phys. Rev. Res., 2, 23–68 (2020).

[5] G. Haller and G. Yuan, Physica D: Nonlinear Phenomena, 147, 352-370 (2000)

[6] G. Haller, Journal of the Mechanics and Physics of Solids, 86, 70–93 (2015).

[7] G. Haller, A. Hadjighasem, M. Farazmand, F. Huhn, Journal of Fluid Mechanics, 795,

136–173 (2016).

[8] C. Schneide, P.P. Vieweg, J. Schumacher, K. Padberg-Gehle, Chaos, 32, 013123 (2022).

[9] R. Teed and H. Latter, MNRAS, 507, 5523-5541 (2021).