Team Report on integration of FSAM to SWMF and on FSAM simulations of convective dynamo and emerging flux in the solar convective envelope

Yuhong Fan and Fang FangNational Center for Atmospheric Research

Code Development and Integration of FSAM into SWMF• Convert and optimize the FSAM code

– From original F77 to F90 version, compatible with SWMF.– Improve the efficiency in message passing and adopt routines from the SWMF.– Integrate FSAM into SWMF as its CZ component following standards and convections of other existing

components such as EE• FSAM can run as a stand-alone mode

– Multiple test cases are available in SWMF/CZ/FSAM/src/problems:• Test run from a solar convection zone background with seed fields: make test • Convective dynamo run from seed fields or from a restart file:

make prob PROBS=dynamo RUNDIR=rundirnamemake prob_rst PROBS=dynamo RUNDIR=rundirname

• A run of isolated rising flux tube in stable cz:make prob PROBS=risetube_qcz RUNDIR=rundirname

• A hydro run for comparison with ASH:make prob PROBS=benchA RUNDIR=rundirname

• New problems can be created by adding new SWMF/CZ/FSAM/src/problems/newprobdir with the necessary files following previous examples

• FSAM can run as a CZ component of SWMF– Wrapper file available in SWMF/CZ/FSAM/srcInterface– Can be set by Config.pl –v=CZ/FSAM– Routine test available in SWMF: make test_cz

• Finite-difference Spherical Anelastic MHD (FSAM) code solves the following anelastic MHD equations in a partial spherical shell domain:

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

A convective dynamo simulation

•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.

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∇• Frad( ) =∇ •16σ sT0

3

3κρ 0

∇T0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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at the bottom ∂s1

∂r= 0, and a latitudinal gradient is imposed :

∂s1

∂θ ⎛ ⎝ ⎜

⎞ ⎠ ⎟=

dsi(θ )dθ

where si θ( ) = −Δsi cosθ − π /2

Δθπ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

at the top s1 is held fixed, and s1 is symmetric at the θ - boundaries

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• Simulation domain r∈ 0.722Rs, 0.971Rs( ), θ ∈ π /2 − π /3, π /2 + π /3( ), φ∈ 0,2π( )• Grid : 96 × 512 × 768, horizontal res. at top boundary 2.8 Mm to 5.5 Mm, vertical res. 1.8 Mm

• K = 3 ×1013 cm2s−1, ν =1012 cm2s−1, η =1012 cm2s−1, at top and all decrease with depth as 1/ ρ

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Ω =2.7 ×10−6rad/s

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Hrad =4π r2

S(r)16σ sT0

3

3κρ 0θ 1

θ 2∫0

2π∫ ∇T0 r2dθ dφ

Hconv =4π r2

S(r)ρ 0θ 1

θ 2∫0

2π∫ T0 v1 s1 r2dθ dφ

Hcond =4π r2

S(r)Kρ 0θ 1

θ 2∫0

2π∫ T0∂s1

∂rr2dθ dφ

where

S(r) = r2dθ dφθ 1

θ 2∫0

2π∫

Consistent with Model S of JCD: the entropy gradient at 0.97Rs is ~10-5 erg g-1 K-1 cm-1

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Re = urms /νk f ≈ 130 to 50 from bottom to top

Co = 2Ω /urms,all k f ≈1.3, where k f = 2π /(ro − ri)

Compared to Re = 36, Co = 7.6 in Kapyla et al. (2012)Ro = urms,all /Ωl ≈ 0.74, where l = H p at bottom

Mean flow properties

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ρ0vp =∇ ×F r,θ( )rsinθ

ˆ φ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

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Hk = v • ∇ × v( )

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Bp =∇ ×A r,θ( )rsinθ

ˆ φ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

maximum

minimum

B=0

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ρ0rsinθ vr⊥vφ

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−rsinθ Br⊥Bφ

RS

RS

MS

Gastine et al. (2014)

• Transition between solar-like and anti-solar differential rotation takes place at Ro ~ 1:

• Presence of the magnetic field suppresses convection, making the flow more rotationally constrained, and changes the Reynolds stress transport of angular momentum:

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Ro = urms,all /ΩH p =0.74 dynamo case0.96 hydro case ⎧ ⎨ ⎩

Examples of strong flux emergence events

• Conform to Hale’s rule by 2.4 to 1 in area• Tilt angle 7.5◦ ± 1.6◦

Summary• MHD simulation of rotating solar convection driven by the radiative heat flux in a

partial spherical shell produces a large-scale mean magnetic field that exhibits irregular cyclic behavior with oscillation time scales ranging from about 5 to 15 years and undergoes irregular polarity reversals

• The mean axisymmetric toroidal magnetic field peaks at the bottom of the convection zone, reaching a value of about 7000 G. Including the fluctuating component, individual channels of strong field reaching 30kG are present.

• The presence of the magnetic fields appears to be important for maintaining the solar-like differential rotation. The magnetic fields make the convective flows more rotationally constrained and the resulting Reynolds stress produces a more outward transport of angular momentum needed for driving a solar-like differential rotation.

• In the midst of magneto-convection, we found occasional emergence of strong super-equipartition flux tubes near the surface, exhibiting properties that are similar to emerging solar active regions.