Nonlinear evolution of unstable solar inertial modes: The case of viscous modes on a differentially rotating sphere

arXiv (Cornell University) · 2026-03-09

On the Sun, the inertial mode with the largest observed amplitude (rms velocity exceeding $10$ m/s) is the high-latitude mode with longitudinal wavenumber $m=1$. In two dimensions, on the sphere, linear theory predicts that this mode is unstable due to a shear instability associated with latitudinal differential rotation (fast equator, slower polar regions). We investigate the evolution of this instability numerically and theoretically. The nonlinear vorticity equation is solved using direct numerical simulations in the time domain. The only control parameter is the Ekman number $E$. For $10^{-3}\lesssim E< E_c \approx 1.5\times10^{-3}$, only the high-latitude $m=1$ mode is unstable. We extract its saturation amplitude as a function of $E$ and compare the results with predictions from two perturbative approaches in nonlinear stability theory. The simulations reveal a supercritical Hopf bifurcation. Near onset, the mode amplitude is well described by the Landau equation $d|A|/dt=σ_I |A|+β_I |A|^3$, with a positive linear growth rate $σ_I$ and a negative nonlinear coefficient $β_I$. The coefficient $β_I$ depends weakly on $E$, implying that the saturated amplitude scales approximately as $|A|\proptoσ_I^{1/2}$. The equilibrium mode contains the $m=1$ fundamental and harmonics $m=2$ and $m=3$, whose amplitudes scale as $σ_I^{m/2}$. Saturation results from Reynolds stresses that smooth the latitudinal differential rotation. For $E=4\times10^{-4}$, consistent with solar-like turbulent viscosity, the saturated velocity reaches $28$ m/s, comparable to solar observations. These results should be interpreted cautiously, since in three dimensions the instability is baroclinic and involves different physics.

Optimal three-dimensional perturbations in fluttering and non-fluttering bioprosthetic aortic valves

Journal of Fluid Mechanics · 2026-03-25

This study examines the transition to turbulence downstream of fluttering and non-fluttering bioprosthetic aortic valves using global linear stability theory. During systole, increasing inflow velocities result in temporally evolving flow profiles downstream of the valve which are highly influenced by the leaflet kinematics. These profiles are time averaged at the sinotubular junction over successive windows and used as boundary conditions to obtain base flows for stability analysis. Three-dimensional global modes are computed for one design of each valve type across multiple time windows, revealing several unstable modes whose frequencies and growth rates increase over time. Notably, the non-fluttering valve exhibits higher growth rates than the fluttering valve. The resulting eigenspectra show that, for each case, the most unstable eigenvalues align along two distinct parabolic branches in the complex plane. For each valve case, the modes within each branch are found to have similar group velocities, suggesting that the unstable modes along a branch constitute a coherent structure. Motivated by this, a transient growth analysis is conducted to identify the optimal initial perturbations that maximise energy gain for a given time horizon. When superimposed onto the base flow, these perturbations generate vortical structures that closely resemble those observed in fully coupled nonlinear fluid–structure interaction simulations for a similar time scale as the one used to obtain the optimal perturbations. These results suggest that the optimal perturbations may initiate the shear-layer instabilities responsible for transition to turbulence, providing valuable insight into the underlying mechanisms in the flow fields downstream of bioprosthetic valve designs.

Learning dissipation and instability fields from chaotic dynamics

Physica D Nonlinear Phenomena · 2025-08-05

HFNO: an interpretable data-driven decomposition strategy for turbulent flows

ArXiv.org · 2025-11-03

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

Data-driven control of fluid flows

Elsevier eBooks · 2025-01-01

Kinematic decomposition of volumetric particle tracking velocimetry data

Experiments in Fluids · 2025-05-22 · 1 citations

Improved Greedy Identification of latent dynamics with application to fluid flows

Computer Methods in Applied Mechanics and Engineering · 2025-02-05 · 1 citations

Mori–Zwanzig latent space Koopman closure for nonlinear autoencoder

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2025-05-01 · 5 citations

The Koopman operator presents an attractive approach to achieve global linearization of nonlinear systems, making it a valuable method for simplifying the understanding of complex dynamics. While data-driven methodologies have exhibited promise in approximating finite Koopman operators, they grapple with various challenges, such as the judicious selection of observables, dimensionality reduction and the ability to predict complex system behaviours accurately. This study presents a novel approach termed Mori–Zwanzig autoencoder (MZ-AE) to robustly approximate the Koopman operator in low-dimensional spaces. The proposed method leverages a nonlinear autoencoder to extract key observables for approximating a finite invariant Koopman subspace and integrates a non-Markovian correction mechanism using the Mori–Zwanzig formalism. Consequently, this approach yields an approximate closure of the dynamics within the latent manifold of the nonlinear autoencoder, thereby enhancing the accuracy and stability of the Koopman operator approximation. Demonstrations showcase the technique’s improved predictive capability for flow around a cylinder. It also provides a low-dimensional approximation for the Kuramoto–Sivashinsky (KS) system with promising short-term predictability and robust long-term statistical performance. By bridging the gap between data-driven techniques and the mathematical foundations of Koopman theory, MZ-AE offers a promising avenue for improved understanding and prediction of complex nonlinear dynamics.

Complex-network modeling of reversal events in two-dimensional turbulent thermal convection

Journal of Fluid Mechanics · 2025-05-13 · 1 citations

Spontaneous flow reversals in buoyancy-driven flows are ubiquitous in many fields of science and engineering, often characterized by violent, intermittent occurrences. In this study, we present a complex-network-based reduced-order model to analyse intermittent events in turbulent flows, using temporal and spatial snapshot data. This framework combines elements of dynamical system theory with network science. We demonstrate its utility by applying it to data sequences from intermittent flow reversal events in two-dimensional thermal convection. This approach has proven robust in detecting and quantifying structures and predicting reversals. Additionally, it provides a perspective on the physical mechanisms underlying flow reversals through cluster evolution. This purely data-driven methodology shows the potential to enhance our understanding, prediction and control of turbulent flows and complex systems.

Fourier-based proper orthogonal decomposition of a turbulent round jet

Journal of Fluid Mechanics · 2025-11-17

We use scanning-tomographic particle image velocimetry introduced by Casey, Sakakibara &amp; Thoroddsen ( Phys. Fluids, vol. 25 (2), 2013, p. 025102) to measure the volumetric velocity field in a fully turbulent round jet. The experiments are performed for ${Re}=2640,\, 5280$ and $10\,700.$ Using Fourier-based proper orthogonal decomposition (POD), the dominant modes that describe the velocity and vorticity fields are extracted. We employ a new method of averaging POD modes from different experimental runs using a phase-synchronisation with respect to a common basis. For the dominant azimuthal wavenumber $m=1,$ the first and second POD modes of the axial velocity have opposite signs and appear as embracing helical structures, with opposite twist, while, for the same parameters, POD modes of the radial velocity extend to the axis of symmetry and, interestingly, also show a helical shape. The $(m=1)$ -POD modes for the azimuthal vorticity appear as two separate structures, consisting of C-shaped loops in the region away from the axis and helically twisted axial tubes close to the axis. The corresponding axial vorticity modes are cone-like and appear as inclined streaks of alternate sign in the $r$ – $z$ -plane, similar to velocity streaks seen in wall-bounded shear flows. Temporal analysis of the dynamics shows that a $(m=1)$ two-mode velocity POD representation precesses with a Strouhal number of approximately $St=0.05,$ while the same reconstruction based on vorticity POD modes has a slightly higher Strouhal number of $St=0.06.$