# Optimized boundary driven flows for dynamos in a sphere

## Abstract

We perform numerical optimization of the axisymmetric flows in a sphere to minimize the critical magnetic Reynolds number Rm{sub cr} required for dynamo onset. The optimization is done for the class of laminar incompressible flows of von Karman type satisfying the steady-state Navier-Stokes equation. Such flows are determined by equatorially antisymmetric profiles of driving azimuthal (toroidal) velocity specified at the spherical boundary. The model is relevant to the Madison plasma dynamo experiment, whose spherical boundary is capable of differential driving of plasma in the azimuthal direction. We show that the dynamo onset in this system depends strongly on details of the driving velocity profile and the fluid Reynolds number Re. It is found that the overall lowest Rm{sub cr} Almost-Equal-To 200 is achieved at Re Almost-Equal-To 240 for the flow, which is hydrodynamically marginally stable. We also show that the optimized flows can sustain dynamos only in the range Rm{sub cr}

- Authors:

- Center for Magnetic Self Organization in Laboratory and Astrophysical Plasmas, University of Wisconsin-Madison, 1150 University Avenue, Madison, Wisconsin 53706 (United States)

- Publication Date:

- OSTI Identifier:
- 22072595

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 19; Journal Issue: 11; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1070-664X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; AXIAL SYMMETRY; BOUNDARY LAYERS; INCOMPRESSIBLE FLOW; LAMINAR FLOW; MAGNETIC REYNOLDS NUMBER; NAVIER-STOKES EQUATIONS; NUMERICAL ANALYSIS; OPTIMIZATION; PLASMA; SPHERES; SPHERICAL CONFIGURATION; STEADY-STATE CONDITIONS; VELOCITY

### Citation Formats

```
Khalzov, I. V., Brown, B. P., Cooper, C. M., Weisberg, D. B., and Forest, C. B.
```*Optimized boundary driven flows for dynamos in a sphere*. United States: N. p., 2012.
Web. doi:10.1063/1.4764048.

```
Khalzov, I. V., Brown, B. P., Cooper, C. M., Weisberg, D. B., & Forest, C. B.
```*Optimized boundary driven flows for dynamos in a sphere*. United States. doi:10.1063/1.4764048.

```
Khalzov, I. V., Brown, B. P., Cooper, C. M., Weisberg, D. B., and Forest, C. B. Thu .
"Optimized boundary driven flows for dynamos in a sphere". United States. doi:10.1063/1.4764048.
```

```
@article{osti_22072595,
```

title = {Optimized boundary driven flows for dynamos in a sphere},

author = {Khalzov, I. V. and Brown, B. P. and Cooper, C. M. and Weisberg, D. B. and Forest, C. B.},

abstractNote = {We perform numerical optimization of the axisymmetric flows in a sphere to minimize the critical magnetic Reynolds number Rm{sub cr} required for dynamo onset. The optimization is done for the class of laminar incompressible flows of von Karman type satisfying the steady-state Navier-Stokes equation. Such flows are determined by equatorially antisymmetric profiles of driving azimuthal (toroidal) velocity specified at the spherical boundary. The model is relevant to the Madison plasma dynamo experiment, whose spherical boundary is capable of differential driving of plasma in the azimuthal direction. We show that the dynamo onset in this system depends strongly on details of the driving velocity profile and the fluid Reynolds number Re. It is found that the overall lowest Rm{sub cr} Almost-Equal-To 200 is achieved at Re Almost-Equal-To 240 for the flow, which is hydrodynamically marginally stable. We also show that the optimized flows can sustain dynamos only in the range Rm{sub cr}},

doi = {10.1063/1.4764048},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 11,

volume = 19,

place = {United States},

year = {2012},

month = {11}

}