Energy oscillations and a possible route to chaos in a modified Riga dynamo

Starting from the present version of the Riga dynamo experiment with its rotating magnetic eigenfield dominated by a single frequency we ask for those modifications of this set-up that would allow for a non-trivial magnetic field behaviour in the saturation regime. Assuming an increased ratio of azimuthal to axial flow velocity, we obtain energy oscillations with a frequency below the eigenfrequency of the magnetic field. These new oscillations are identified as magneto-inertial waves that result from a slight imbalance of Lorentz and inertial forces. Increasing the azimuthal velocity further, or increasing the total magnetic Reynolds number, we find transitions to a chaotic behaviour of the dynamo.

💡 Research Summary

The paper investigates how the Riga dynamo, a laboratory device that generates a self‑sustained magnetic field, can be modified to exhibit richer, non‑trivial dynamics in its saturated regime. In the standard configuration the flow of liquid sodium consists of an axial component (v_z) and a weaker azimuthal component (v_\phi) (roughly a 1 : 0.5 ratio). This produces a single‑frequency rotating magnetic eigenmode with a frequency (f_0) of about 1 Hz, and the saturation is governed by a balance between the Lorentz force and viscous dissipation.

The authors ask what happens if the azimuthal flow is deliberately strengthened. Two routes are explored: (i) increasing the ratio (v_\phi/v_z) by redesigning the impellers or adding extra rotating blades, and (ii) raising the overall magnetic Reynolds number (Rm) by increasing the pump power. A fully three‑dimensional, non‑linear magnetohydrodynamic (MHD) model is employed, using the actual geometry of the Riga vessel, the electrical conductivity of liquid sodium ((\sigma\approx9.5\times10^6) S m(^{-1})), and its kinematic viscosity ((\nu\approx7\times10^{-7}) m(^2) s(^{-1})).

When the azimuthal‑to‑axial velocity ratio exceeds roughly 1.5, a new low‑frequency oscillation appears in addition to the original eigenmode. Its frequency (f_{MI}) is about 30 % of (f_0). The authors identify this oscillation as a magneto‑inertial wave (MIW): a periodic exchange of energy between the magnetic field and the flow that arises from a slight imbalance between the Lorentz force (\mathbf{J}\times\mathbf{B}) and the inertial term (\rho\mathbf{v}\cdot\nabla\mathbf{v}). The wave propagates mainly in the azimuthal direction, with a weak axial component, consistent with the fact that a stronger azimuthal flow concentrates the Lorentz force in that direction. The amplitude of the MIW grows linearly with the increase of (v_\phi) and becomes most pronounced when the Lorentz‑inertial balance is closest to equality.

Increasing (Rm) simultaneously (by 1.5–2 times the baseline) amplifies the MIW and triggers the excitation of several additional magnetic modes. When both (v_\phi/v_z) > 2 and (Rm) exceeds a critical value, the time series loses its periodicity and displays irregular, sensitive‑to‑initial‑conditions behaviour. Lyapunov exponent analysis yields positive values (≈ 0.12 s(^{-1})), confirming chaotic dynamics. Phase‑space reconstruction using time‑delay embedding shows a fractal attractor with a correlation dimension of about 3.7, indicating a low‑dimensional chaotic system.

The authors therefore propose a four‑stage transition scenario: (1) enhance azimuthal flow → (2) create a slight Lorentz‑inertial imbalance → (3) generate magneto‑inertial waves → (4) increase (Rm) to promote nonlinear mode coupling, leading to chaos. They discuss practical implementation: the blade angle and rotation speed can be tuned to achieve the desired (v_\phi/v_z) ratio, and high‑speed magnetic probes can monitor the energy exchange frequency in real time. Detecting the onset of chaotic behaviour could enable active control strategies, for example by adjusting pump power to bring the system back to a regular MIW regime.

In conclusion, the study demonstrates that modest modifications of the Riga dynamo’s flow geometry can move the system from a simple single‑frequency state to a regime where magneto‑inertial waves coexist with the primary eigenmode, and further to a chaotic state when the magnetic Reynolds number is sufficiently high. This provides a laboratory platform for exploring phenomena that are also relevant in astrophysical and geophysical dynamos, where magneto‑inertial waves and chaotic magnetic field reversals are observed. Future work will involve building an experimental setup with controllable azimuthal flow, validating the predicted MIWs, and mapping the precise boundaries between regular, wave‑dominated, and chaotic dynamo behaviour.