Physics	  of	  Flux	  Emergence	  on	  the	  SunMark Cheung
Staff Physicist @ Lockheed Martin Solar & Astrophysics Laboratory
Visiting Associate Professor @ National Astronomical Observatory of Japan

Plan•

Lecture I: Physics of Flux Emergence•Lecture II: Diagnostics of Flux EmergenceAims of this talk/article• a primer on the physics that govern the behaviorof magnetic fields emerging into the solar
atmosphere,• an introduction of how flux emergence plays akey role in many aspects of solar physics, and• a broad overview of established and more recent
developments in this field of research.

Science questions•

What is the mechanism that brings the magnetic flux fromthe interior to the atmosphere? (see Yuhong Fan’s Living
Reviews article)•How does emerging flux transport the magnetic energy
and helicity?

•

How much flux is trapped below the surface during flux
emergence, and what is the contribution of this trapped
flux to the solar dynamo?

•

What are the roles of emerging flux in the free energy
accumulation and triggering of the transient events such as
jets, flares, CMEs?

Science questions•

What are the physical properties of subsurface magneticstructures that rise and eventually emerge onto thesurface?

•Does flux emergence occur as the rise of coherent bundles
or as smaller elementary units? (Zwaan, 1978, 1985)?

•What is the difference between flare-productive sunspots
and quiet sunspots?

•How do convective flows impact the morphology andphysical character of emerging flux (e.g., Fan et al., 2003;
Cheung et al., 2007a)?

Science questions•

How do the individual and statistical properties (e.g.,
Hagenaar, 2001; Hagenaar et al., 2003; Iida et al., 2012; Otsujiet al., 2011) of emerging flux relate to the solar activity cycle?

•What are the observational consequences of emerging
magnetic flux (Bruzek, 1967; Zwaan, 1978), and what are thephysical mechanisms responsible?

•Can we predict the appearance of new emerging flux regions?

•What are the physical ingredients necessary for a realistic
model of emerging flux?

10 (30) arc seconds

Hinode SOT observation trilobite (Credit: Title)(Credit: Yang Liu, Stanford)

Hinode SOT observation of sunspot formation (Shimizu,Ichimoto & Suematsu 2012): A Ca II dark ring precedes
appearance of the penumbra in G-band BFI images

Hinode SOT observation of sunspot formation (Shimizu,
Ichimoto & Suematsu 2012): A Ca II dark ring precedes
appearance of the penumbra in G-band BFI images

A fluid view of MHD
Magnetohydrodynamics (MHD) captures thefollowing physical principles:•

Mass conservation,•Momentum conservation,•Energy conservation,•Faraday’s law of induction. (next slide)Faraday’s Induction Equation

Fluid expansion / compression

Stretching flow along
magnetic field lines
intensifies B

Faraday’s Induction Equation

Fluid expansion / compression

Stretching flow along
magnetic field lines
intensifies B

Constitutive Relations
Equation of Statep = p(ϱ,ε), s = s(ϱ,ε),T = T(ϱ,ε)

Ohm’s Law cE = -vxB + η∇xB
Diffusion coefficients 𝞂, Q = ∇·(𝜅∇T), ηRadiative properties Q = -∇·(Frad)
Constitutive relations are statements about the material propertiesof the medium (in this case, plasma). The choice of different
constitutive relations spawns a diversity of MHD models.

A Diversity of MHD Models

Magne&c(Field((Dominated(Plasma(Pressure(Dominated(Martinez-Sykora et al. (2008, 2009)

Buoyancy

Magnetic Buoyancy (Parker 1955)

Spruit et al. (1987)
Consider a blob of plasma threaded by B, which
exhibits magnetic pressure B2/8π. When the blob isin pressure balance with its surroundings,

pin + B2/8π = pext.
If the plasma blob has the same temperature or
specific entropy as the surrounding (i.e. Tin = Text, or
sin = sext), then it will have a density deficit, andhence will rise like an air bubble.

Surface

0.0000001 g/cm3

Gravitational Stratification•The solar convection zone is highly stratified.

Depth = 200,000 km

0.1 g/cm3

Surface

Depth = 200,000 km Toriumi & Yokoyama (2011)

0.0000001 g/cm3

0.1 g/cm3

Scaling Relation Between B and 𝜚(a) Horizontal tube
expanding as it rises
B 𝜚∝(b) Crest of tube
expanding predominantlyin the horizontal directions
B 𝜚1/2∝

Flux Emergence Workshop 2011 @ SSL

Inversion Layer

Cheung et al (2007)

Inversion Layer

PhotosphereBuoyancy Instabilities

Matsumoto et al. (1993)

Non-linear
developmentof the undular(Parker) mode

Shibata et al. (1989)

Magara (2001)

Isobe et al (2007)

Convection

Hinode/SOT observation of an emerging flux region

Vertical velocity structure at various depths (Stein et al 2011)

Surface Intensification of Emerging Magnetic Field

From Cheung et al. 2008: When a magnetic parcel reaches
the photosphere and radiatively cools, its density andmagnetic field strength enhances to give an inversion layer.
The relative enhancement for |B| is greater than for massdensity due to field stretching.

Tortosa-Andreu & Moreno-Insertis (2009)

Lites (2009): Hinode/SP scan of an emerging flux region

Radiative MHD simulation of AR formation (Cheung et al. 2010)

Red$(nega)ve$magne)c$ﬂux)$$$$$Green$(horizontal$magne)c$ﬁeld)$$$Blue$(posi)ve$magne)c$ﬂux)$
Purple$background$(brightness$intensity$of$convec)ve$ﬂows)$

Mass discharge
Courtesy: Stein

Schematic
Top: Dark blue indicates high density
scenario in 2D
Bottom: Red indicates downflow
From Cheung, Rempel, Title & Schuessler (2010). Samemechanism as suggested by Kubo, Low & Lites (2010).

Rempel & Cheung (2014)

ɸ dot x4ɸ dot x4

ɸ dot x4

Flux Spreading in the Decay Phase
Decay due to turbulent
diffusion (see Meyer et al1974 and Mosher 1977)

Green dashed lines show theself-similar solution for:
Φ0 = 1.1 x 1022 Mx,
σ0 = 11 Mm, andηTurb = 350 km2s-1.

Figure 9. Transport of photospheric (z = 0) magnetic flux for the leading polarity
contours
showthesurfacesof constantenclosed
within aYellow
circle ofradius r fromaxis leading
spot. The yellow
contours correspto ⇥ =flux
1022inMx
in incrementsof 21⇥ Mx= 2 (for1021
Green dashedlines in this pa
intervalsof 2x10theMx.
leadingpolarity).
solution for the decay of an ideal spot due to turbulent di⇤usion. The remaining(negative) value at radial distance r at a certain time t indicating that the flux ⇥ eConvection-driven Emergence

Isobe, Proctor & Weiss (2008)•

Magnetoconvection brings small-scale loops up to thesurface, which then reconnect with the surrounding field.
This is a source of high-frequency waves in the atmosphere.

TwistTwist

Fan et al. (1999): Non-linear development of the helical kinkinstability in a buoyant flux tube. See Takasao’s talk for helicallykink unstable emerging flux.

Fan (2009): Distribution of vorticity (verticalcomponent) at the photosphere from a simulation of
the emergence of a twisted flux rope. The two ‘spots’
have the same sign of rotation.

AManchester et al. (2004)

Consider a closed loop about a sunspot. Take thepath integral of the force. The contributions fromthe pressure terms vanish, but a torque remains
from the Lorentz force. So any systematic twist in
the underlying magnetic structure will drive net
rotational flows (e.g. Sturrock, Hood & Archontis2015).

A Simple Model by Longcope & Welsch (2000)•“Current shunting” model for
twisted active region emergence•

Idealized emerging active region
has net twist•

Matched to force-free coronal field•

At the interface (photosphere), a
horizontally diverging current
drives a torque, which sends a
torsional Alfvén wave down thetube.

•

Over an Alfvén crossing time ~ 1day (100 Mm @ 1 km/s), the tube
unwinds while the coronal field istwisted up.

From Magara & Longcope (2003): Poynting fluxes of magnetic energy andrelative helicity in a 3D flux emergence simulation. Note how the shear(horizontal motion) contribution outlasts the emergence (vertical motion)
contribution.

Origin of Twist?
Still somewhat controversial

Mean-field Dynamo 𝝰 effect*? Longcope et al. (1999) says it predicts thewrong sign of twist.
• They propose instead the 𝝨 Effect: namely an initial flux tube in the solar
interior has zero magnetic helicity. Interaction with helical convection
introduces writhe of tube axis. Helicity conservation means twist of✿
opposite sign is produced .
• What about wrapping of poloidal field lines over a rising flux tube
(Choudhuri, Chatterjee & Nandy 2004)?
• Studies of the solar cycle variation of active region twist may put important
constraints to reject models. See work of Hagino & Sakurai, M. Zhang,
Pevtsov and collaborators. For latest, consult the Space Science Reviews
paper by Pevtsov, Berger, Nindos, Norton & van Driel-Gesztelyi (2015).
• Global convective dynamo simulations (e.g. Hotta, Nelson, Fan) that selfconsistently generate flux tubes may provide an answer (or tell us it’smuch more complicated).
*The 𝝰 effect is an EMF parallel to <B> due to turbulence.
✿ But but but … Seehafer (Phys Rev E, 1283, 1996): The 𝝰 effectgenerates helicity in the small and large scales with equal amplitude butopposite signs!
•Ion-neutral interaction
Hi, I’m an ion.

Ion-neutral interaction
Hi, I’m an ion.
Are you sure?

Ion-neutral interaction
Hi, I’m an ion.
Are you sure?
I’m positive!

Generalized Ohm’s Law

Physical effects of Ohm’s Law
Fluid Motion
Induction

OhmicDiffusion

Hall EffectAmbipolarDiffusion

Of the form-E=v*xB?

Yes / uNo

Yes / jYes / jxB

Changes
topology?

NoYesNoNoDissipative?

NoYesNoYesLeake & Arber (2006): Inclusion of the Cowling resistivity (i.e.
Ohmic + ambipolar) leads to increased rate of flux emergence(right). The emerged field is also more force-free.
However, they used a simplified 1D prescription for theionization degree of hydrogen. For (more) self-consistent
simulations investigating the effect the ambipolar effect in thesolar atmosphere, refer to Juan’s papers. (Time-dependent Hionization?)

Interaction with pre-existing fieldSDO/AIA

Interaction with pre-existing fieldYokoyama & Shibata (1996): 2D MHD
simulation of current sheet formation,
leading to plasmoid ejections

Nishizuka et al. (2008):
Alfvén wave generation in jet simulation

Moreno-Insertis et al. (2008):
3D jet simulation from flux emergence into a coronal holeMiyagoshi & Yokoyama (2003)

Left: with anisotropic thermal conduction
Right: without anisotropic conduction

Data-Driven Modeling

Chen et al. (2014): Simulation of AR corona formation. The
simulation includes anisotropic thermal conduction and is driven
at the bottom boundary by a model from Rempel & Cheung(2014).

What we need for real data-driven models
Retrieval of electric fields. E-fields of interest at the photosphere cannot
be directly measured with remote sensing. It rises from a combination ofeffects:• -v x B• Non-ideal parallel E-fields• Mean-field EMFs arising from correlations at small scales ( - <v’xB’> )• Finite spatio-temporal resolution and sampling (gives rise toaveraging)
Example problems:a. Retrieve the E-field needed for driving (surface) flux transportmodels to study the solar cycle (dynamo)
b. Retrieve the E-field integrated along the solar equatorc. To measure the full Poynting flux for magnetic energy and helicity
Hi-fidelity, validated methods for reliable retrieval of photospheric driving
E-fields are needed. This is in the works thanks to a number of efforts
involving Fisher, Kazachenko, Welsch, Fan, Rempel, Cheung, Linton,Yeates, Peter, Feng; this is not an exhaustive list).

Electric field and Poynting Flux
The formula for the time evolution of Hrel (Berger & Field, 1984; Demoulin &Berger (2003) has many terms. Usually only two terms (blue + red) areincluded by helicity practitioners.
Volumetricdissipation term

Photospheric flux due tohorizontal motions (a.k.a.
shearing term)

Flux through nonphotospheric boundary

Photospheric flux due to verticalmotions (a.k.a. emergence term)vxB should really be replaced by -E (e.g. Kazachenko et al.
2014).

Data-Driven Modeling with Magnetofriction• Balance of Lorentz force and fictitious frictional force (Yang,Sturrock & Antiochos, 1986; Craig & Sneyd 1986)– Plasma velocity proportional to Lorentz force:v = 𝝂-1 jxB where 𝝂 is the frictional coefficient– Evolve magnetic field according to Induction Equation• Total magnetic energy in volume monotonically decreasing(provided net Poynting flux through boundaries is zero).

Data-Driven Modeling with Magnetofriction• Van Ballegooijen, Priest & Mackay (2000)–Evolve vector potential A–Plasma velocity proportional to Lorentz
force: v = (𝝂0 B2)-1 jxB• Yeates, Mackay & Van Ballegooijen (2008)–Global magnetofrictional model ofcoronal field in response to observedchanges in photospheric field,including• Differential rotation, meridional
circulation• Flux dispersal and cancellation• Appearance of AR-scale, twisted
bipoles–Correctly reproduces filament chiralityYeates, Mackay & Van Ballegooijen 2008
and location
See also Yeates 2013
–Memory of corona ~ 6 weeks to a fewmonths

Credit: Keiji Hayashi

Data-Driven model of AR 11158• CGEM* project: Fisher et al. 2015, Space Weather, 13.
• Electric field boundary condition from inversions byUC Berkeley SSL group (see Kazachenko et al. 2014)
with support from Stanford group (Sun, Liu &Hoeksema).
• Magnetofriction model (Cheung & DeRosa) of theevolution of NOAA AR 11158 over 5+ days.

*CGEM is funded by the NASA/NSF Strategic Capability program

Magnetograms at different heights

Visualization of field lines based on current density• There is no impulsive eruption at the time of the observed X-flare. However,moments before this time, a current-carrying flux rope is observed to form
and is eventually ejected, though the rise time is on the order of hours.

• An extension of this work is to use the destabilized configuration as the initialcondition for a MHD run using RADMHD (in progress by Abbett & Bercik).

A Diversity of MHD Models

Thank you
for yourattention

For much more
please refer toCheung & Isobe(2014)