Low-dimensional stochastic amplitude equations for a precessing rotating cylinder

The magnetic field of planets and stars is generated by the movement of conductive fluids inside these bodies. The precession and libration of these astrophysical bodies play a central role in the excitation of the internal turbulent fluid motion. In our laboratory, we have developed an experiment that allows the investigation of precession-driven inertial waves and their instability (Xu and Harlander, 2020). Wave triads play a very important role in this instability (Lagrange et al., 2011). As the Ekman number decreases, an increasing number of interacting triads arise, ultimately leading to turbulence. This process can be experimentally reproduced in the laboratory. In this experiment, precession is simulated using a slightly tilted cavity with a free fluid surface and is therefore simpler in design than a real precession experiment. 

The dynamics of fluids can be described by PDEs. However, often deeper insights can be gained from a corresponding low-dimensional dynamical system. An example is the large family of Lorenz-type models, which have led to a fundamental understanding of predictability in atmospheric dynamics (Majda et al., 1999). Also, for the problem of a precessing rotating cylinder, low-dimensional models exist. Such models are obtained from spectral discretizations of the Navier-Stokes equations and truncating the resulting hierarchy of coupled equations at low order. Truncation, however, eliminates the quadratic coupling between the resolved modes and the (unresolved) smaller scales, which can lead to unrealistic characteristics of turbulence. 

We suggest another closure to systematically derive low-order amplitude equations for rotating fluids, based on stochastic modeling of the unresolved small scales in accordance with the available experimental data. Specifically, we first remodel the small scales by an appropriate stochastic process that has a multivariate Gaussian law when conditioned on the resolved variables and, in a second step, apply a projection operator to the coupled system. In doing so, we derive closed, averaged equations for the resolved variables that retain the quadratic nonlinearities and so capture the small-scale contributions to the low-order wave dynamics. For a projection operator in the form of a conditional expectation (i.e., a projection on function space), we have recently studied necessary and sufficient conditions under which the projection operator formalism yields an approximation for nonreversible (e.g. driven) systems (Duong et al., 2025). Measuring the distance between the marginal distributions of the resolved variables for the full- and the low-order models, the accuracy of the low-order model can be measured (Hartmann et al., 2020).  

By comparing the low-order stochastic model results with data from the precession experiment, the hope is not only to capture the wave interactions correctly and develop a stochastic extension of the existing amplitude equations, but also to reduce the order of the existing model even further. 

M.H Duong, C. Hartmann, and M. Ottobre, arXiv preprint,  arXiv:2506.14939, 2025.

C. Hartmann, L. Neureither, and U. Sharma, SIAM J. Math. Anal. 52(3), 2689-2733, 2020.

R. Lagrange, P. Meunier, F. Nadal, C Eloy, J. Fluids Mech. 666, 104–145, 2011.

A.J. Majda, I. Tomofeyev, E. Vanden Eijnden, PNAS, 96(26), 14687-14691, 1999.

W. Xu, U. Harlander, Rev. Phys. Fluids., 5(9), 094801-21, 2020.