The High Altitude Observatory (HAO) at the National Center for Atmospheric Research (NCAR).

The National Center for Atmospheric Research is sponsored by the National Science Foundation. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Differential Rotation and Emerging

Flux in Solar Convective Dynamo

Simulations

Yuhong Fan (HAO/NCAR), Fang Fang (LASP/CU)

GTP workshop August 17, 2016

• Finite-difference Spherical Anelastic MHD (FSAM) code solves the following

anelastic MHD equations in a partial spherical shell domain:

 

where D is the viscous stress tensor, and Frad is the radiative diffusive heat flux :

· Simulation domain r Î 0.722Rs, 0.971Rs( ), q Î p / 2 - p / 3, p / 2 + p / 3( ), f Î 0,2p( )

· Grid: 96(r)´ 512(q )´ 768(f), horizontal res. at top boundary 2.8 Mm to 5.5 Mm, vertical res. 1.8 Mm

· K = 3´1013 cm2s-1, n =1012 cm2s-1, h =1012 cm2s-1, at top and decrease with depth as 1/ r (Fan and Fang 2014)

· K is the same as above but n=0, h=0 (Fan and Fang 2016)

Simulation setup

• Convection is driven by the radiative diffusive heat flux as a source term in the entropy equation:

• The boundary condition for s1:

• The velocity boundary condition is non-penetrating and stress free at the and the top, bottom and q-

bondaries

• For the magnetic field: perfect conducting walls for the bottom and the q-boundaries; radial field at the top

boundary

• Angular rotation rate for the reference frame , net angular momentum relative to the

reference frame is zero.

Ñ×Frad

Fan and Fang (2014)

at r = 0.73 Rs

maximum

minimum

dynamo

HD

HVHD

B=0

increase

viscosity

 

RS = r0r̂ ¢ v r̂ ¢ v f MS = -1

4pr̂ Br̂ Bf

Ro = urms / (WH p) = 0.74

Ro = urms / (WH p) = 0.96

Ro = urms / (WH p) = 0.71

Angular momentum flux density due to meridional circulation:

Fmc = r0L vm , where L = r̂2 W

If L » L(r̂ ), then the net angular momentum flux across

a constant r̂ cylinder (e.g. Miesch 2005, Rempel 2005):

f (r̂ ) = r0L(r̂ )s( r̂ )

ò vm dS » L(r̂ ) r0

s( r̂ )

ò vm dS = 0

• Meridional circulation does not transport net angular momentum

across the cylinders, but only redistributes it along the cylinders

• A net angular momentum flux across the cylinders by the Reynolds

stress cannot be balanced by meridional circulation differential

rotation

dynamo

HD

HVHD

B=0

increase

viscosity

 

RS = r0r̂ ¢ v r̂ ¢ v f MS = -1

4pr̂ Br̂ Bf

Ro = urms / (WH p) = 0.74

Ro = urms / (WH p) = 0.96

Ro = urms / (WH p) = 0.71

dynamo HVHD

HD

Less diffusive dynamo dynamo

Less diffusive dynamo

dynamo

m=0 modes excluded

Solid lines: less diffusive dynamo

Dashed lines: dynamo

Dotted line: hydro

Solid lines: less diffusive dynamo

Dashed lines: dynamo

Dotted line: hydro

Solid lines: less diffusive dynamo

Dashed lines: dynamo

Less diffusive

dynamo

dynamo

Wbuoyancy 0.782 Lsol 0.777 Lsol

Wlorentz 0.330 Lsol 0.256 Lsol

with latitudinal gradient

of s at base of CZ

without latitudinal gradient

of s at base of CZ

•Conform to Hale’s rule by 2.4 to 1 in area

• Statistically significant mean tilt angle: 7.5◦ ± 1.6◦

•Weak Joy’s law trend

• Tilt angles of super-equipartition flux emergence areas at 0.957 Rs:

Strong emerging flux bundles sheared by giant cell convection

• Emerging fields are not isolated flux tubes rising from the bottom of the CZ, but are continually formed

in the bulk of the CZ through sheared amplification by the giant-cell convection

• The emerging flux bundle is sheared into a hairpin shape with the leading end up against the down flow

lane of the giant cell.

• Inserting strong, buoyant toroidal flux tubes at the base of the CZ with the

background convective dynamo:

Untwisted toroidal

flux tubes:

B = 100 kG,

a = 0.16 Hp,

F = 2.5 × 1023

Mx

B >H p

a

æ

èç

ö

ø÷

1/2

Beq

Beq for rms speed ~ 5-10 kG

Beq for peak downflow ~ 60 kG

For magnetic buoyancy

to dominate convection:

• Tilt angles of super-equipartition flux emergence areas at 0.957 Rs:

Less diffusive dynamo case:

•Conform to Hale’s rule by 2.7 to 1 in area

• Statistically significant mean tilt angle: 4.5◦ ± 1.6◦

•Weak Joy’s law trend

Less diffusive dynamo + 2 strong toroidal tubes:

•Conform to Hale’s rule by 2.7 to 1 in area

• Statistically significant mean tilt angle: 7.4◦ ± 2.2◦

•Weak Joy’s law trend

• Inserting strong toroidal flux tubes at the base of the CZ with the background

convective dynamo:

Untwisted toroidal

flux tubes:

B = 100 kG,

a = 0.24 Hp,

F = 5.6 × 1023

Mx

Summary

• Simulations of 3D convective dynamo in the rotating solar convective envelope driven by the

radiative heat flux:

o produce a large-scale mean magnetic field that exhibits irregular cyclic behavior with

polarity reversals.

o Produce emergence of coherent super-equipartition toroidal flux bundles with properties

similar to solar active regions: following Hale’s rule by 2.7:1, a weak Joy’s law trend.

o The presence of the magnetic fields are necessary for the self-consistent maintenance of

the solar-like differential rotation. In several ways it acts like an enhanced viscosity.

o With further reduced magnetic diffusivity and viscosity, we find an enhanced magnetic

energy, and a reduced kinetic energy in the large scales. This results in a further

enhanced outward transport (away from the rotation axis) of angular momentum by the

Reynolds stress, balanced by an increased inward transport by the magnetic stress, with

the transport by the viscous stress reduced to a negligible level

• Simulations of buoyant rise of highly super-equipartition toroidal flux tubes from the bottom of

the CZ in the background convective dynamo:

o For B ≤ 100 kG, the rise of active region scale flux tubes are strongly influenced by the

giant-cell convection and become in-distinguishable from the background dynamo fields

after rising to about the middle of the CZ

o The rise of larger flux tubes (significantly greater than active region flux) from the

bottom of CZ may result in changes to the differential rotation that are incompatible with

observations.