Abstract
The results of a numerical study of the magnetic dynamo effect in cylindrical von Kármán plasma flow are presented with parameters relevant to the Madison Plasma Couette Experiment. This experiment is designed to investigate a broad class of phenomena in flowing plasmas. In a plasma, the magnetic Prandtl number can be of order unity (i.e., the fluidReynolds number is comparable to the magnetic Reynolds number). This is in contrast to liquid metal experiments, where is small (so, ) and the flows are always turbulent. We explore dynamo action through simulations using the extended magnetohydrodynamic NIMROD code for an isothermal and compressible plasma model. We also study twofluid effects in simulations by including the Hall term in Ohm’s law. We find that the counterrotating von Kármán flow results in sustained dynamo action and the selfgeneration of magnetic field when the magnetic Reynolds number exceeds a critical value. For the plasma parameters of the experiment, this field saturates at an amplitude corresponding to a new stable equilibrium (a laminar dynamo). We show that compressibility in the plasma results in an increase of the critical magnetic Reynolds number, while inclusion of the Hall term in Ohm’s law changes the amplitude of the saturated dynamo field but not the critical value for the onset of dynamo action.
The authors wish to thank Dr. C. Sovinec for valuable help and discussions related to NIMROD. The work is supported by the National Science Foundation.
I. INTRODUCTION
II. NIMROD MODELS OF MPCX
III. HYDRODYNAMICAL EQUILIBRIUM AND STABILITY
IV. KINEMATIC DYNAMO
V. NONLINEAR SATURATION OF DYNAMO FIELD
VI. DISCUSSION
Key Topics
 Magnetic fields
 34.0
 Magnetohydrodynamics
 26.0
 Plasma flows
 24.0
 Reynolds stress modeling
 19.0
 Laminar flows
 12.0
Figures
Madison Plasma Couette Experiment (MPCX): (a) sketch; (b) partial vertical cross section. Rings of permanent magnets of alternating polarity line the inside of the cylinder with their poles oriented normally to the walls. Ring anodes and cathodes are placed between the magnets. The resulting drift is in the azimuthal direction. By varying the potential between the anodes and cathodes, the velocity forcing at the outer boundary can be customized.
Madison Plasma Couette Experiment (MPCX): (a) sketch; (b) partial vertical cross section. Rings of permanent magnets of alternating polarity line the inside of the cylinder with their poles oriented normally to the walls. Ring anodes and cathodes are placed between the magnets. The resulting drift is in the azimuthal direction. By varying the potential between the anodes and cathodes, the velocity forcing at the outer boundary can be customized.
Structure of axisymmetric equilibrium von Kármán flow driven by electromagnetic system at the boundary for Mach number , fluid Reynolds number , and magnetic Reynolds number : (a) number density; (b) velocity; (c) magnetic field. Crosssections in plane are given. Left panels of (b) and (c) show stream lines of poloidal parts ( and components) of flux and magnetic field , respectively, superimposed on absolute values of these parts depicted in colors. Right panels of (b) and (c) show azimuthal components of corresponding fields.
Structure of axisymmetric equilibrium von Kármán flow driven by electromagnetic system at the boundary for Mach number , fluid Reynolds number , and magnetic Reynolds number : (a) number density; (b) velocity; (c) magnetic field. Crosssections in plane are given. Left panels of (b) and (c) show stream lines of poloidal parts ( and components) of flux and magnetic field , respectively, superimposed on absolute values of these parts depicted in colors. Right panels of (b) and (c) show azimuthal components of corresponding fields.
Structure of axisymmetric equilibrium von Kármán flow driven by differentially rotating walls for Mach number and fluid Reynolds number : (a) number density; (b) velocity. The same elements as in Fig. 2 are shown.
Structure of axisymmetric equilibrium von Kármán flow driven by differentially rotating walls for Mach number and fluid Reynolds number : (a) number density; (b) velocity. The same elements as in Fig. 2 are shown.
Time dynamics of kinetic energy of different azimuthal harmonics in purely hydrodynamical von Kármán flow for Mach number and fluid Reynolds number . Corresponding azimuthal mode numbers are shown.
Time dynamics of kinetic energy of different azimuthal harmonics in purely hydrodynamical von Kármán flow for Mach number and fluid Reynolds number . Corresponding azimuthal mode numbers are shown.
Kinetic energy of different azimuthal harmonics in hydrodynamically stable von Kármán flow as a function of (a) fluid Reynolds number for Mach number ; (b) Mach number for fluid Reynolds number . Corresponding azimuthal mode numbers are shown.
Kinetic energy of different azimuthal harmonics in hydrodynamically stable von Kármán flow as a function of (a) fluid Reynolds number for Mach number ; (b) Mach number for fluid Reynolds number . Corresponding azimuthal mode numbers are shown.
Critical magnetic Reynolds number as a function of (a) fluid Reynolds number for Mach number ; (b) Mach number for fluid Reynolds number . Vertical line in (a) separates the regions of axisymmetric and nonaxisymmetric equilibrium von Kármán flows.
Critical magnetic Reynolds number as a function of (a) fluid Reynolds number for Mach number ; (b) Mach number for fluid Reynolds number . Vertical line in (a) separates the regions of axisymmetric and nonaxisymmetric equilibrium von Kármán flows.
Time dynamics of kinetic and magnetic energies of different azimuthal modes in von Kármán flow for Mach number , fluid Reynolds , and magnetic Reynolds , with azimuthal mode numbers labeled. [(a),(b)] Singlefluid MHD case . Flows are of even modes while fields are odd in . [(c),(d)] Hall MHD case . Initial behavior is similar to the singlefluid MHD case, but as the fields become strong , the Hall effect becomes important. The final equilibrium includes both odd and even .
Time dynamics of kinetic and magnetic energies of different azimuthal modes in von Kármán flow for Mach number , fluid Reynolds , and magnetic Reynolds , with azimuthal mode numbers labeled. [(a),(b)] Singlefluid MHD case . Flows are of even modes while fields are odd in . [(c),(d)] Hall MHD case . Initial behavior is similar to the singlefluid MHD case, but as the fields become strong , the Hall effect becomes important. The final equilibrium includes both odd and even .
Magnetic field lines of saturated dynamos: (a) singlefluid MHD case ; (b) Hall MHD case . Thickness of the line is proportional to the magnitude of the field, while lighter (darker) color corresponds to upward (downward) direction of the field.
Magnetic field lines of saturated dynamos: (a) singlefluid MHD case ; (b) Hall MHD case . Thickness of the line is proportional to the magnitude of the field, while lighter (darker) color corresponds to upward (downward) direction of the field.
Magnetic energy of different azimuthal harmonics in saturated Hall MHD dynamo as a function of the Hall number . Azimuthal mode numbers are shown. Dashed line corresponds to scaling .
Magnetic energy of different azimuthal harmonics in saturated Hall MHD dynamo as a function of the Hall number . Azimuthal mode numbers are shown. Dashed line corresponds to scaling .
Tables
Parameters of MPCX.
Parameters of MPCX.
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