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

€

Re =Ω1r1(r2 − r1)

υ≈107

€

Rm =Ω1r1(r2 − r1)

η≈ 20

€

S =VA (r2 − r1)

η≈ 4

€

r1 = 7.1(cm)

€

r2 = 20.3(cm)

€

Ω1 = 4000(rpm)

€

Ω2 = 533(rpm)

€

Bz = 0.5(T)

€

h = 27.9(cm)

€

M = maxV flow

Vsound

≤1

4

Physical Parameters

€

ρ =6.0(g /cm3)

€

ν =3⋅10−3(cm2 /s)

€

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

€

∝ M 2

€

M ≈1

4

Incompressible Code*

*Ref. A. Kageyama, H. Ji, J. Goodman, F. Chen, and E. Shoshan, J. Phys. Soc. Japan. 73, 2424 (2004)

€

Re =11,600

€

Re = 5,800

€

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 ( )

€

Re = 400,Rm = 400

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

€

ψ

€

φ

€

Bϕ

€

Vϕ

Flux Function Stream Function

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

Rapid Jet Current Sheet

€

max(Vpoloidal

Vtoroidal

) ≈1

6

€

max(Bϕ

Bz

) ≈1

2

Speed and Width of outward jet

From the theory*

€

V jet ∝ Rm−0.5

From the simulation

€

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,

€

Voutside ∝ Rm−0.5

From Mass Conservation

€

V jetW jet = Voutside (2h −W jet )

Thus

€

W jet ∝ Rm0

Z-Average torques versus RRe=400 Rm=400

Initial State Final State

Increase of Total Torque on cylinders

€

Γfinal /Γinitial ∝Re0.22 Rm0

€

(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 ( ).

€

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)

€

V jet ∝ Rm−0.53

€

ψ /φ∝η

€

W jet ∝ Rm0

€

tsaturation ∝ tη ∝η −1 >> tdynamics ∝Ω−1€

Γfinal /Γinitial ∝Re0.22 Rm0

The END

Thanks