Nonmodal and nonlinear dynamics of helical magnetorotational instability

The helical magnetorotational instability (HMRI), a relative of standard MRI (SMRI), has become a subject of active research in recent years in connection with the experiments on magnetized cylindrical Taylor-Couette (TC) flows. It occurs in the presence of helical magnetic field, consisting of azimuthal and axial components and, like SMRI with only axial magnetic field, taps into the rotational energy of the flow. However, a main advantage of HMRI is that, being governed by the Reynolds (Re) and Hartmann (Ha) numbers, it persists even at very small magnetic Prandtl numbers typical to liquid metals, in contrast to SMRI. The linear development of HMRI has been widely studied theoretically using both classical modal and more recently by nonmodal stability analysis, where a fundamental connection between nonmodal dynamics and dissipation-induced (double-diffusive) modal instabilities, such as HMRI, has been demonstrated. A series of specially designed liquid metal TC experiments provided the first experimental evidence of HMRI and reproduced the main results of the linear theory, such as the stability threshold and propagation speed (frequency) of HMRI-wave. More importantly, these experiments revealed much richer dynamics of HMRI as a function of system parameters (Re, Ha, etc.) than that obtained from the linear analysis only. These results prompted further theoretical studies of the nonlinear development of HMRI, but detailed physics of its saturation and sustenance still remains missing, especially when comparison with the experiment is concerned.
Motivated by the existing experimental results, we investigate the evolution of HMRI, from its linear growth to nonlinear saturation using numerical simulations.
We show that depending on the Reynolds number, two regimes of saturation can be realized. At Re below a certain critical value (but higher than the instability threshold), the saturation energy linearly depends on Re and the corresponding energy spectrum is dominated by the most unstable mode and its multiple wavenumbers, while at larger Re, the energy increases with Re, but not linearly, and the related spectrum looks like turbulent spectrum, being much smoother over wavenumbers. The nonlinear state remains markedly axisymmetric (m = 0)
and at high Re can be viewed as a 2D turbulence, whose (spectral) properties are further examined.