Simulations of Princeton Gallium Experiment Wei Liu Jeremy Goodman Hantao Ji Jim Stone Michael J....
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Transcript of Simulations of Princeton Gallium Experiment Wei Liu Jeremy Goodman Hantao Ji Jim Stone Michael J....
Simulations of Princeton Gallium Experiment
Wei Liu
Jeremy Goodman
Hantao Ji
Jim Stone
Michael J. Burin
Ethan Schartman
CMSO
Plasma Physics Laboratory
Princeton University, Princeton NJ, 08543
Research supported by the US Department of Energy, NASA under grant ATP03-0084-0106 and APRA04-0000-0152 and also by the National Science Foundation under grant AST-0205903
Outline
• Introduction
• Linear Simulations of MRI
• Nonlinear Saturation of MRI
• Conclusions
Diagram and Parameters
Control Dimensionless Parameters
• Reynolds Number
• Magnetic Reynolds Number
• Lundquist Number
• Mach Number
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Re =Ω1r1(r2 − r1)
υ≈107
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Rm =Ω1r1(r2 − r1)
η≈ 20
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S =VA (r2 − r1)
η≈ 4
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r1 = 7.1(cm)
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r2 = 20.3(cm)
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Ω1 = 4000(rpm)
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Ω2 = 533(rpm)
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Bz = 0.5(T)
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h = 27.9(cm)
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M = maxV flow
Vsound
≤1
4
Physical Parameters
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ρ =6.0(g /cm3)
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ν =3⋅10−3(cm2 /s)
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η =2 ⋅103(cm2 /s)
Material Properties (Liquid Gallium)
Main Features and modifications of ZEUS
(1) Add viscous term into Euler Equation with azimuthal viscous term in flux-conservation form
(2) Add resistive term to Induction Equation by defining equivalent electromotive force
(3) Boundary condition:
Magnetic Field: Vertically Periodic, Horizontally Conducting
Velocity Field: Vertically Periodic, Horizontally NO-SLIP
(4) Benchmarks against Wendl’s Low Reynolds Number Test** and Magnetic Gauss Diffusion Test
ZEUS: An explicit, compressible astrophysical MHD code*
Modified for non-ideal MHD:
* Ref. J. Stone and M. Norman, ApJS. 80, 753 (1992)
J. Stone and M. Norman, ApJS, 80, 791 (1992)
**Ref. M.C.Wendl, Phys. Rev. E. 60, 6192 (1999)
Comparison with Incompressible CodeRe=1600
Compressible Code
• Low Mach Number ( ) with NO-SLIP boundary condition on cylinders and end-caps
• Error
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∝ M 2
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M ≈1
4
Incompressible Code*
*Ref. A. Kageyama, H. Ji, J. Goodman, F. Chen, and E. Shoshan, J. Phys. Soc. Japan. 73, 2424 (2004)
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Re =11,600
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Re = 5,800
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Re = 3,800
Linear MRI SimulationComparison with Local Linear Analysis*
*Ref. H. Ji, J. Goodman and A. Kageyama, Mon. Not. R. Astron. Soc. 325, L1 (2001)
Linear MRI SimulationComparison with Global Linear Analysis*
Rm Re Vertical Harmonic
Growth Rate (/s)
Global Simulation
400 400 1 41.67 71.66
2 72.71
3 77.69
4 56.88
5 0.283
20 1 23.31 35.52
2 32.61
3 23.73
4 6.905€
"∞"
*Ref. J. Goodman and H. Ji Fluid Mech. 462, 365 (2002)
Nonlinear Saturation
Rotating Speed Profile ( )
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Re = 400,Rm = 400
Nonlinear Saturation (Re=400,Rm=400,M=1/4,S=4)
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ψ
€
φ
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Bϕ
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Vϕ
Flux Function Stream Function
Rapid outward Jet and Current Sheet (Re=400,Rm=400,M=1/4,S=4)
Rapid Jet Current Sheet
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max(Vpoloidal
Vtoroidal
) ≈1
6
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max(Bϕ
Bz
) ≈1
2
Speed and Width of outward jet
From the theory*
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V jet ∝ Rm−0.5
From the simulation
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V jet ∝ Rm−0.53
* Ref. E. Knobloch and K. Julien, Mon. Not. R. Astron. Soc. 000, 1-6 (2005)€
(200 ≤ Rm ≤ 6400)
And roughly,
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Voutside ∝ Rm−0.5
From Mass Conservation
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V jetW jet = Voutside (2h −W jet )
Thus
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W jet ∝ Rm0
Z-Average torques versus RRe=400 Rm=400
Initial State Final State
Increase of Total Torque on cylinders
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Γfinal /Γinitial ∝Re0.22 Rm0
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(200 ≤ Rm ≤ 6400)
Conclusions about Nonlinear SaturationM=1/4,S=4
• At final state, the rotating profile is flattened somewhat, uniform rotation results*.
• The width of the “jet” is almost independent of resistivity, but it does decrease with increasing Re; the speed of the “jet” scales as*:
• At final state the total torque integrated over cylinders depends somewhat upon viscosity but hardly upon resistivity.
• The smaller the resistivity, the longer is required to reach the final state. Oscillations appear to persist indefinitely if Rm>800*.
• The ratio of the poloidal flow speed to the poloidal field strength is proportional to resistivity**.
These conclusions apply at large Rm ( ).
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Rm ≥ 200
* Ref. E. Knobloch and K. Julien, Mon. Not. R. Astron. Soc. 000, 1-6 (2005)
**Ref. F. Militelo and F. Porcelli, Phys. Plasmas 11, L13 (2004)
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V jet ∝ Rm−0.53
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ψ /φ∝η
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W jet ∝ Rm0
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tsaturation ∝ tη ∝η −1 >> tdynamics ∝Ω−1€
Γfinal /Γinitial ∝Re0.22 Rm0
The END
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