The MPDX: (a) a sketch of the experiment; (b) the electrode configuration near the wall for driving plasma velocity . The spherical system of coordinates is shown. Center line (CL) corresponds to the axis of symmetry. Reprinted with permission from Phys. Plasmas 19, 104501 (2012). Copyright 2012 American Institute of Physics.
Profiles of driving boundary velocity for flows I, II, and III.
Equilibrium structures of flows I, II, and III for fluid Reynolds number Re = 150. The left half of each figure shows stream lines of poloidal velocity superimposed on its absolute values depicted in colors, the right half shows a contour plot of azimuthal velocity (dashed curves denote values of ). The vertical central lines represent the axis of symmetry.
Critical magnetic Reynolds as a function of fluid Reynolds Re for flow III.
Ratio of poloidal to toroidal kinetic energies as a function of fluid Reynolds number Re for flows I, II, and III.
Contour plot of critical magnetic Reynolds number as a function of fluid Reynolds number Re and driving parameter . Contours of are shown. The shaded area denotes the hydrodynamically unstable region, with stability boundaries shown for azimuthal modes m = 1, 2, 3 (curves labeled with symbols). Each point of the solid black curve corresponds to the optimized stable flow that minimizes at a given value of Re. Points of the dashed black curve correspond to the optimized flows at the boundary of hydrodynamic stability. Symbol “” denotes the point (), at which the global minimum of is achieved. The segmentation of the dynamo/no dynamo boundary is due to discrete scan of the plane .
Real part of dynamo growth rate as a function of magnetic Reynolds number Rm for fluid Reynolds number Re = 150 and different values of driving parameters (solid curve), (dashed curve), (dashed-dotted curve) and (dotted curve).
Critical magnetic Reynolds as a function of fluid Reynolds Re for optimized flows with different number of driving harmonics. Solid curves correspond to the optimized hydrodynamically stable flows, dashed curves correspond to the optimized flows at the boundary of hydrodynamic stability. N denotes the number of the highest non-zero Fourier harmonic in the driving velocity . Symbol “” denotes the overall lowest value of , which is achieved at .
Dependences of driving Fourier coefficients on fluid Reynolds Re in optimized flows with different number of driving harmonics. Values of Re are scanned with step . As in Fig. 8, solid curves correspond to the stable flows, dashed curves correspond to the marginally stable flows.
Profiles of optimized azimuthal velocity at the boundary for different number of driving harmonics, fluid Reynolds number is Re = 150.
Axisymmetric equilibrium flows corresponding to optimized driving velocities from Fig. 10 at fluid Reynolds number Re = 150. Notations are the same as in Fig. 3.
Dependences of real (solid curves) and imaginary (dashed curves) parts of dynamo growth rate on magnetic Reynolds number Rm for the optimized flows shown in Fig. 11. Calculations are done for the fastest dynamo mode (with azimuthal mode number m = 1).
Magnetic field lines of the fastest kinematic dynamo eigen-modes obtained at Rm = 400 for the optimized flows shown in Fig. 11. Thickness of the lines is proportional to the magnitude of the field. Vertical lines represent the axis of symmetry of the flows. Horizontal lines (shown in darker color) denote the axis of these equatorial dipole-like dynamo fields.
Expected parameters of MPDX. Dimensionless numbers Re, Rm, and Pm are estimated from the Braginskii equations20 (see corresponding formulas in Refs. 10 and 22).
Fourier coefficients of driving velocities for flows I, II, and III.
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