CMSO 2005

obabko@uchicago.edu

Simulation of Gallium experiment*§

Aleksandr Obabko

Center for Magnetic-Self OrganizationDepartment of Astronomy and Astrophysics

University of Chicago

*with Fausto Cattaneo (U of C) and Paul Fischer (ANL)§used NERSC resources supported by the Office of Science of the US DOE under Contract No.

DE-AC03-76SF00098 (INCITE 2005)

CMSO 2005

Problem formulation

PmReRmRRRR

Re

Rmt

pRet

,))((

0

1)()(

2

1)()(

122211

2

22

Bu

BuBBuB

BuBBuu

u

• Incompressible viscous resistive MHD equations in cylindrical geometry of MRI experiment at PPPL

• Boundary conditions:

0 ,

,0

,)( ,0

,0

zo,z

z

21

HzRRr

r

r

z

BBB

BB

rrVVV

• Solved numerically by MHD version of ANL spectral element code nek5000

22

214

2

2113

222

111

,

,

,

,

)(

Rrr

rRr

RrR

RrR

r

RR

RR

CMSO 2005

Overview of Results

• Periodic: Stable Couette flow for Bz0→0 & Bz0→∞, otherwise, MRI (for Rm>100…300)

• Rings diminish Ekman circulation (and torque) as compared to cases with lids & exhibit similar growth of torque with increase of Bz0

Rings setup is more suitable for study of MRI & angular momentum transport in laboratory

Torque / TorqueCouette

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Bz0

Rings Re=6e3 T_i

Rings Re=6e3 T_o

Rings Re=2e3 T_i

Rings Re=2e3 T_o

Periodic Re=2e3

Periodic Re=6e3

Lids Re=2e3 Ti

Lids Re=6e3 Ti

• 2D & 3D hydro & MHD runs with vertical B.C. of periodicity, rings & lids• Re=620 … 6200, Pm=0.5 & Bz0=0 … 0.275

CMSO 2005

2D Steady Hydro Results for Re=620

• Angular momentum transport outward both by outward radial flow of high angular momentum and by inward radial flow of low angular momentum (like convection)

• Rings diminish Ekman circulation and hydro angular momentum transport restored Couette flow profile

zrF

rV

Lids: Rings:

V

Couette flow

Lids

Rings

Lids

Rings

zrF

CMSO 2005

2D Unsteady Results for Re=6200 with rings

• Smaller viscous boundary & shear layers• Oscillating vertical jets • Short radial jet outward at midplane of z=H/2

V

Couette flow

zrFzrF zrF

rV rV

CMSO 2005

2D MHD: Re=6200, Rm=3100, B0=0.05, rings

• The reversal of radial flow (toward the inner cylinder along the plane at z=H/2)• Higher torques compared to hydro case• Hydro angular momentum transport near boundaries, magnetic – near z=H/2

V

Couette flow

zrF

zrFrV zrMzrH

rV

rB

zrH zrMrB zrF

Total

Hydro

Magnetic

CMSO 2005

3D Results for Re=6200

• Initial conditions: 2D flow solutions• Weak 3D dimensionality in hydro cases

Similar angular momentum transport• Smaller structures compared to 2D cases

tzrF zr ,,, tzrF zr ,,

2D

Hydro with rings

3D

tzrF zr ,,, tzrF zr ,,

Periodic MHD, Bz0=0.05

2D 3D

CMSO 2005

Conclusion & Future Work

Conducted 2D & 3D Hydro & 2D MHD computations with vertical B.C. of periodicity, rings & lids in geometrical setup of Princeton MRI experiment

Observed diminished Ekman circulation and angular momentum transport in hydro cases with rings as compared to the cases with lids

Found similar behavior of torque with increase of external axial magnetic field Bz0 for 2D MHD cases with rings as compared to the periodic case with MRI

Suitability of the ring setup for study of MRI and angular momentum transport in laboratory

Angular momentum transport is reminiscent of transport of thermal energy in convection

…Ongoing 3D MHD calculations

…Plans for high-Re (~106) 2D MHD Case