3D Spherical Shell Simulations of Rising Flux Tubes in the Solar

Convective Envelope

Yuhong Fan (HAO/NCAR)

High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR)

The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Researchunder sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.

Outline

• Overview of results from the thin flux tube

model and from MHD simulations in local

Cartesian geometries

• New results from simulations of buoyantly

rising magnetic flux tubes in the solar

convective envelope using a spherical shell

anelastic MHD code.

Figure by George Fisher

Full disk magnetogram from KPNO

The Thin Flux Tube Model

• Thin flux tube approximation: all physical quantities are averages over the tube cross-

section, solve for the mean motion of each tube segment under the relevant forces:

• Results:– Field strength of the toroidal magnetic field at the base of SCZ is

of order

– Tilt of the emerging loop: active region tilts, Joy’s law– Asymmetric inclination of the two sides of the emerging loop– Asymmetric field strength between the two sides of the loop

G510

,pHa

For 2D horizontal tubes: twist rate , where (e.g. Moreno-Insertis & Emonet 1996, Fan et al. 1998; Longcope et al. 1999).

For 3D arched flux tubes: necessary twist may be less, depending on the initial conditions (e.g. Abbett et al. 2000; Fan 2001)…

MHD Simulations in Local Cartesian Geometries

• The dynamic effects of field-line twist:

– Maintaining cohesion of rising flux tubes

untwisted

twisted

untwisted twisted

2/1)( aHq p zrBBq /

Fan et al. (1998) Abbett et al. (2001)

Anelastic MHD Simulations in a Spherical Shell

Anelastic MHD Simulations in a Spherical Shell

We solve the above anelastic MHD equations in a spherical shell representing the solar convective envelope (which may include a sub-adiabatically stratified stable thin overshoot layer):

— staggered finite-difference

— two-step predictor-corrector time stepping

— An upwind, monotonicity-preserving interpolation scheme is used for evaluating the fluxes of the advection terms in the momentum equations

— A method of characteristics that is upwind in the Alfven waves is used for evaluating the V x B term in the induction equation (Stone & Norman 1992).

— The constrained transport scheme is used for advancing the induction equation to ensure that B remains divergence free.

— Solving the elliptic equation for at every sub-time step to ensure

• FFT in the -direction a 2D linear system for each azimuthal order

• The 2D linear equation (in ) for each azimuthal order is solved with the generalized cyclic reduction scheme of Swartztrauber (NCAR’s FISHPACK).

1p 0)( 0 v

,r mm

Anelastic MHD Simulations in a Spherical Shell

.153.0,10 150

aqGB

axisymmetriccentral cross-section of -tube

axisymmetric

const.sin)sin( rrV

• In the axisymmetric case, the angular momentum of each tube segment is conserved.

central cross-section of -tube

-tube

axisymmetic

13.0 aq

13.0 aq

2.9tilt

3.0 1

aq

7.4tilt

3.0 1

aq

• A twisted flux tube when arched upward will rotate out of the plane, i.e. develop a writhe.

• For a left-hand-twisted (right-hand twisted) tube, the rotation is counter-clockwise (clockwise) when viewed from the top.

.1515.0,10 150

aqGB

Apex cross section

.153.0,10 150

aqGB

.1515.0,10 150

aqGB

For the emergence of a left-hand-twisted flux tube, the polarity orientation starts out as south-north oriented, and then after an apparent shearing motion, establishes the correct tilt.

.150,1050

qGB

Summary

From the simulations of the buoyant rise of -shaped flux tubes in a rotating spherical model solar convective envelope, it is found:

• The rise trajectory for a 3D -tube is more radial than that of an axisymmetric toroidal flux ring.

• A twisted flux tube when arched upward develops a tilt that is counter-clockwise (clockwise) when viewed from the top if the twist is left-handed (right-heanded). Since flux tubes in the northern hemisphere are preferentially left-hand twisted, the twist is driving a tilt opposite to the effect of the Coriolis force and opposite to the direction of the observed mean active region tilt. We find that in order for the buoyant flux tube to emerge with a tilt consistent with observations, the twist of the flux tube needs to be less than half of the critical twist necessary for the tube to rise cohesively. Under such conditions, severe flux loss ( > 50% of the total flux) is expected during the rise.

Summary (cont.)

• Due to the asymmetric stretching of the rising -tube by the Coriolis force, a field strength asymmetry develops with the leading side of the emerging tube being greater in field strength and more cohesive compared to the following side. This provides a natural explanation of the observe morphological asymmetry of solar active regions where the leading polarity of an active region tends to be more cohesive, usually in the form of a large sunspot, while the following polarity tends to appear more fragmented.

• A retrograde flow of about 100m/s is present in the apex segment of the rising -tube. This may be a deep signature to look for to detect rising active region flux tubes prior to their emergence?

Future work• Self-consistently model the formation and rise of buoyant flux tubes from the base of the solar convection zone:

– What are the instabilities that can lead to the formation of active region scale flux tubes, e.g. magnetic buoyancy instabilities (modified by solar rotation)?

– What determines the twist of the magnetic flux tubes that form, given the current helicity of the magnetic fields generated by the dynamo?

• Incorporating convection into the simulations:

– Is convection important in determining the properties of emerging active region flux tubes?