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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

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Outline Introduction Linear Simulations of MRI Nonlinear Saturation of MRI Conclusions

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Diagram and Parameters Control Dimensionless Parameters Reynolds Number Magnetic Reynolds Number Lundquist Number Mach Number Physical Parameters Material Properties (Liquid Gallium)

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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 Wendls 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)

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Wendls Low Reynolds Number Limit Test (Re=1,Rm=0,M=1/4,S=0) Azimuthal Velocity distribution along r direction With NO-SLIP boundary condition on cylinders and end- caps* *Ref. M.C.Wendl, Phys. Rev. E. 60, 6192 (1999)

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Gauss Magnetic Diffusion Test Magnetic Reynolds Number (Rm) Resolution N x N Simulated Decay Rate (1/s) Theoretical Results (1/s) Relative Error (%) 400100x100382.52642392.260482.482 40050x50352.76963391.874549.979 100100x1001533.64601569.04192.256 10050x501420.40781567.49829.384 The error scales quadratically with cell size, as expected for our second-order difference schemes

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Comparison with Incompressible Code Re=1600 Compressible Code Low Mach Number ( ) with NO-SLIP boundary condition on cylinders and end-caps Error 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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Linear MRI Simulation Comparison with Local Linear Analysis* *Ref. H. Ji, J. Goodman and A. Kageyama, Mon. Not. R. Astron. Soc. 325, L1 (2001)

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Linear MRI Simulation Comparison with Global Linear Analysis* RmReVertical Harmonic Growth Rate (/s) GlobalSimulation 400 141.6771.66 272.71 377.69 456.88 50.283 20123.3135.52 232.61 323.73 46.905 *Ref. J. Goodman and H. Ji Fluid Mech. 462, 365 (2002)

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Linear MRI Simulation Time evolution for various initial modes

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Nonlinear Saturation (M=1/4,S=4) The conclusions are only valid for large Rm ( ), though I have studied the cases with ( ) and ( ). I will give some typical results with modest Re and modest Rm.

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Nonlinear Saturation Rotating Speed Profile ( )

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Nonlinear Saturation (Re=400,Rm=400,M=1/4,S=4) Flux FunctionStream Function

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Rapid outward Jet and Current Sheet (Re=400,Rm=400,M=1/4,S=4) Rapid JetCurrent Sheet

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Speed and Width of outward jet From the theory* From the simulation * Ref. E. Knobloch and K. Julien, Mon. Not. R. Astron. Soc. 000, 1-6 (2005) And roughly, From Mass Conservation Thus

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Z-Average torques versus R Re=400 Rm=400 Initial State Final State

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Increase of Total Torque on cylinders

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Conclusions about Nonlinear Saturation M=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 ( ). * 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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Other Simulations Different Boundary condition* Magnetic Field: Vertically Periodic, Horizontally Insulating Velocity Field: Vertically Periodic, Horizontally NO-SLIP Helical MRI simulation** Current-Free initial toroidal magnetic fields Horizontally Conducting and Vertically Insulating Boundary Condition 1.Magnetic Ekman Circulation - Ekman-Hartman Layer*** 2.Magnetic Stewartson Layer with split end-caps Full Insulating Boundary Condition Simulate the experiment as real as possible * Ref. J. Goodman and H. Ji Fluid Mech. 462, 365 (2002) ** Ref. R. Hollerbach and G. Rudiger, PRL 95, 124501 (2005) ***Ref. P. Gilman and E. Benten, Phys. Fluids 11, 2397 (1968)

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Ekman Layer Thickness

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Diagram of Magnetic Ekman Circulation Simulation Regime A: Liguid Gallium Regime B: Vacuum Regime C: Conducting Red line: Conducting Boundary Blue line: Insulating Boundary Differential rotating with initial vertical Magnetic Field

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Magnetized Ekman Circulation Magnetic Interaction parameter: it measures the relative importance of the magnetic field. The width of the Ekman layer decreases with increasing the strength of axial magnetic field. Hartmann Current exerts a strong influence on the outer flow

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Stream Function and Poloidal Current Profile

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Magnetic Stewartson Layer with split end-caps (Re=3200, Rm=2) Estimate M scaling with reasonable experimental parameters

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Magnetic Ekman & Stewartson Layer 4 conclusions With rings, Stewartson layer has more influence than Ekman layer to the system With low Reynolds number, the system will become stable eventually. With high Reynolds number, the system will become turbulent eventually. The higher Reynolds number is, the more deviated the azimuthally speed profile is form ideal Couette state. This means the Stewartson layer extends deeper into the fluid. The stronger the magnetic field is, the more deviated the azimuthally speed profile is from ideal Couette state. This means the Stewartson layer extends deeper into the fluid Strong magnetic field will suppress turbulence. When magnetic strength is large enough, the system becomes stable eventually. This means strong magnetic field causes Stewartson layer stable

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The END Thanks

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