Nonlinear instability of a wide-gap spherical Couette flow in the presence of weak noise

  • Krivonosova, Olga (Lomonosov Moscow state university)
  • Gritsevich, Maria (Helsinki University)
  • Ivanov, Oleg (Lomonosov Moscow state university)
  • Zhilenko, Dmitry (Lomonosov Moscow state university)

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The flow of a viscous incompressible fluid between the rotating inner sphere and fixed outer one is under consideration. The spherical gap is equal in width to the inner sphere radius. We study numerically the transition between supercritical flow to subcritical one. White noise with equal excitation of all frequencies is introduced into the flow by adding fluctuations, completely chaotic in time and with zero mean, to the rotational rate of the inner sphere. The studies conducted are important for understanding the effect of time-dependent rotational rates of large-scale geophysical flows such as those in the atmosphere and mantle of the Earth. Mean flow generation under noise action was found to occur larger for the subcritical flows as compared to the flows after first instability. Similar generation of mean flows is well studied in the case of periodic in time oscillations of rotational rates (see Koch at al. (2013), Cebron at al. (2021)). A slight decrease in the critical Reynolds number, corresponding to the onset of the first instability in the form of travelling azimuthal waves, was found under the action of additional weak noise. This result was obtained from the Reynolds number dependences of oscillation amplitudes, root mean square deviations and mean values of azimuthal velocities for Reynolds numbers exceeding the critical value. Three-dimensional calculations in this case are very expensive. Although the dependence of the flow velocity oscillation amplitudes on Re number in noise free case is well-known (Landau, Lifshitz (1987)) it is unknown in the presence of noise (Lissandrello et al. (2015)). We have proposed a new approach to simplify the three-dimensional calculations of the flow stability limit location in the presence of noise. This approach based on analysis of the exponential damping of the oscillations amplitudes at the Re numbers less than the critical value. We have demonstrated the possibility of reduction of the calculation time steps by using the proposed method. The results obtained by foregoing methods and with proposed approach, demonstrate good agreement. We suppose that this new approach will be useful also for another time-dependent cases, for example, for periodic oscillations of the rotational rate. This may be the subject for further research. This study was carried out with the funding of a grant from the Russian Science Foundation, project 23-29-00051